Which expression is equivalent to (3x^2 + 5x - 2) + (x^2 - 7x + 6) ?
Answer choices Hint Combine terms with the same variable and the same exponent. Remember that x^2 means 1x^2 .
Step-by-step solution Group the like terms: (3x^2 + x^2) + (5x - 7x) + (-2 + 6) . x^2 -terms: 3x^2 + 1x^2 = 4x^2 (the exponent stays 2).x -terms: 5x - 7x = -2x .Constants: -2 + 6 = 4 . The sum is 4x^2 - 2x + 4
Check: let x = 2 . Original: (12 + 10 - 2) + (4 - 14 + 6) = 20 + (-4) = 16 . Answer: 16 - 4 + 4 = 16 ✓
Tip: adding like terms is like adding apples: 3x^2 + x^2 is 4 of the same thing, 4x^2 . Exponents never change when you add.
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Which expression is equivalent to (5x^2 - 3x + 4) - (2x^2 - 6x - 1) ?
Answer choices Hint The minus sign in front of the second parentheses changes the sign of every term inside.
Step-by-step solution Distribute the minus sign: -(2x^2 - 6x - 1) = -2x^2 + 6x + 1 . Write it all out: 5x^2 - 3x + 4 - 2x^2 + 6x + 1 . x^2 -terms: 5x^2 - 2x^2 = 3x^2 .x -terms: -3x + 6x = 3x .Constants: 4 + 1 = 5 . The difference is 3x^2 + 3x + 5
Check: let x = 2 . Original: (20 - 6 + 4) - (8 - 12 - 1) = 18 - (-5) = 23 . Answer: 12 + 6 + 5 = 23 ✓
Tip: rewrite subtraction as "add the opposite" before combining: flip every sign in the second polynomial, then add.
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Type your answer in the box. Fractions like 3/4 are fine.
Hint Use FOIL: First, Outer, Inner, Last. Then combine the two middle terms.
Step-by-step solution First: x \cdot x = x^2 . Outer: x \cdot (-2) = -2x . Inner: 6 \cdot x = 6x . Last: 6 \cdot (-2) = -12 . Combine the middle terms: -2x + 6x = 4x . So (x + 6)(x - 2) = x^2 + 4x - 12 , which gives b = 4 c = -12
Check: let x = 3 . Original: (9)(1) = 9 . Answer: 9 + 12 - 12 = 9 ✓
Tip: for (x + p)(x + q) , the middle coefficient is p + q and the constant is p \times q . Here 6 + (-2) = 4 and 6 \times (-2) = -12 .
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Add the polynomials 4x^3 - 2x + 7 and -x^3 + 5x^2 - 9 . What is the sum, written in descending order of exponents?
Answer choices Hint Line up terms with the same variable and the same exponent. A term written as -x^3 has a coefficient of -1 .
Step-by-step solution Remove the parentheses (adding does not change any signs): 4x^3 - 2x + 7 - x^3 + 5x^2 - 9 . Group like terms by exponent: (4x^3 - x^3) + 5x^2 + (-2x) + (7 - 9) . x^3 -terms: 4x^3 - 1x^3 = 3x^3 .x^2 -terms: only 5x^2 (the first polynomial has no x^2 -term).x -terms: only -2x .Constants: 7 - 9 = -2 . The sum is 3x^3 + 5x^2 - 2x - 2
Check: let x = 1 . Original: (4 - 2 + 7) + (-1 + 5 - 9) = 9 + (-5) = 4 . Answer: 3 + 5 - 2 - 2 = 4 ✓
Tip: write the answer in descending powers (x^3 , x^2 , x , number). If a power appears in only one polynomial, it simply carries down unchanged.
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Type your answer in the box. Fractions like 3/4 are fine.
Hint "Total" means add the two expressions. Combine the n^2 -terms, then the n -terms, then the numbers.
Step-by-step solution Total = (2n^2 + 9n - 14) + (5n^2 - 4n + 30) . n^2 -terms: 2n^2 + 5n^2 = 7n^2 , so a = 7 .n -terms: 9n + (-4n) = 5n , so b = 5 .Constants: -14 + 30 = 16 , so c = 16 . Total daily sales: 7n^2 + 5n + 16 , so a = 7 , b = 5 , c = 16
Check: let n = 2 . Downtown: 8 + 18 - 14 = 12 . Riverside: 20 - 8 + 30 = 42 . Sum = 54 . Answer: 7(4) + 5(2) + 16 = 28 + 10 + 16 = 54 ✓
Tip: words like total , combined and altogether signal addition. Keep each sign attached to the term after it: the -4n stays negative when you add.
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Simplify the difference below. Which choice shows the result in standard form?
(7x^2 + 2x - 5) - (3x^2 - 4x + 8)
Answer choices Hint The minus sign in front of the second parentheses changes the sign of every term inside it.
Step-by-step solution Distribute the minus sign to each term of (3x^2 - 4x + 8) : it becomes -3x^2 + 4x - 8 . Rewrite: 7x^2 + 2x - 5 - 3x^2 + 4x - 8 . x^2 -terms: 7x^2 - 3x^2 = 4x^2 .x -terms: 2x + 4x = 6x .Constants: -5 - 8 = -13 . The difference is 4x^2 + 6x - 13
Check: let x = 1 . Original: (7 + 2 - 5) - (3 - 4 + 8) = 4 - 7 = -3 . Answer: 4 + 6 - 13 = -3 ✓
Tip: before combining anything, rewrite the subtracted polynomial with all its signs flipped. Then the problem is just addition, and you will not lose a sign halfway through.
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Distribute and simplify. What is the product of the monomial -4x^2 and the trinomial 3x^2 - 5x + 2 ?
Answer choices Hint Multiply -4x^2 by each of the three terms. Multiply the coefficients (watch the signs) and add the exponents.
Step-by-step solution First term: (-4x^2)(3x^2) = (-4 \cdot 3)x^{2+2} = -12x^4 . Second term: (-4x^2)(-5x) = (-4)(-5)x^{2+1} = +20x^3 (negative times negative is positive). Third term: (-4x^2)(2) = -8x^2 . Put the terms together: -12x^4 + 20x^3 - 8x^2 . The product is -12x^4 + 20x^3 - 8x^2
Check: let x = 1 . Original: -4(1)(3 - 5 + 2) = -4(0) = 0 . Answer: -12 + 20 - 8 = 0 ✓. Try x = -1 too: original -4(1)(3 + 5 + 2) = -40 ; answer -12 - 20 - 8 = -40 ✓
Tip: count the minus signs in each product. One negative factor gives a negative result; two negative factors give a positive result.
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A tiled wall has (3x - 4) rows of tiles, with (x + 5) tiles in each row. Which polynomial gives the total number of tiles on the wall?
Answer choices Hint Use FOIL: First, Outer, Inner, Last. Then combine the two middle terms.
Step-by-step solution Total tiles = rows \times tiles per row = (3x - 4)(x + 5) . Multiply with FOIL:
First: 3x \cdot x = 3x^2 .Outer: 3x \cdot 5 = 15x .Inner: -4 \cdot x = -4x .Last: -4 \cdot 5 = -20 .Combine the middle terms: 15x - 4x = 11x . The product is 3x^2 + 11x - 20
Check: let x = 2 . Original: (6 - 4)(2 + 5) = 2 \cdot 7 = 14 . Answer: 3(4) + 11(2) - 20 = 12 + 22 - 20 = 14 ✓
Tip: a product of two binomials usually has three terms after combining. If your answer has only two terms, check whether you skipped the Outer and Inner products.
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Which expression is equivalent to (2x - 3)^2 ?
Answer choices Hint Squaring means multiplying by itself: (2x - 3)(2x - 3) . Do all four products.
Step-by-step solution Write it as a product: (2x - 3)(2x - 3) . First: 2x \cdot 2x = 4x^2 . Outer: 2x \cdot (-3) = -6x . Inner: -3 \cdot 2x = -6x . Last: (-3)(-3) = 9 . Combine: 4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9 . The answer is 4x^2 - 12x + 9
Check: with the pattern (a - b)^2 = a^2 - 2ab + b^2 : a = 2x , b = 3 gives 4x^2 - 2(2x)(3) + 9 = 4x^2 - 12x + 9 ✓. With x = 2 : (1)^2 = 1 and 16 - 24 + 9 = 1 ✓
Tip: a squared binomial always has three terms, and the middle term is twice the product of the two terms.
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Which expression is equivalent to 3x(2x^2 - 4x + 5) ?
Answer choices Hint Multiply 3x by each of the three terms. When you multiply powers of x , add the exponents.
Step-by-step solution 3x \cdot 2x^2 = 6x^{1 + 2} = 6x^3 .3x \cdot (-4x) = -12x^{1 + 1} = -12x^2 .3x \cdot 5 = 15x .Put the terms together: 6x^3 - 12x^2 + 15x . The answer is 6x^3 - 12x^2 + 15x
Check: let x = 2 . Original: 6(8 - 8 + 5) = 6(5) = 30 . Answer: 48 - 48 + 30 = 30 ✓
Tip: every term gets multiplied by the number and the variable outside. A quick count: the answer has as many terms as the parentheses had (here, 3), and each power of x goes up by one.
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Which expression is equivalent to \dfrac{x^2 + 7x + 12}{x + 3} for x \ne -3 ?
Answer choices Hint Factor the numerator. Find two numbers that multiply to 12 and add to 7.
Step-by-step solution Factor x^2 + 7x + 12 : we need two numbers with product 12 and sum 7. Those are 3 and 4. So x^2 + 7x + 12 = (x + 3)(x + 4) . Divide: \dfrac{(x + 3)(x + 4)}{x + 3} . The common factor x + 3 cancels. What is left is x + 4 . The answer is x + 4
Check: multiply back: (x + 4)(x + 3) = x^2 + 3x + 4x + 12 = x^2 + 7x + 12 ✓. With x = 2 : \dfrac{4 + 14 + 12}{5} = \dfrac{30}{5} = 6 , and 2 + 4 = 6 ✓
Tip: you can check any division answer by multiplying it by the divisor. If you get the original numerator, it is right.
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A rectangular rug is (2x + 5) feet long and (x - 1) feet wide. Which expression represents the area of the rug, in square feet?
Answer choices Hint Area of a rectangle = length × width (on the formula sheet). Multiply the two binomials with FOIL.
Step-by-step solution Area: A = lw = (2x + 5)(x - 1) . First: 2x \cdot x = 2x^2 . Outer: 2x \cdot (-1) = -2x . Inner: 5 \cdot x = 5x . Last: 5 \cdot (-1) = -5 . Combine: 2x^2 - 2x + 5x - 5 = 2x^2 + 3x - 5 . The area is 2x^2 + 3x - 5
Check: let x = 3 . The rug is 11 ft by 2 ft, so its area is 22 ft². The expression gives 18 + 9 - 5 = 22 ✓
Tip: area is a product, so the answer has an x^2 term. An answer with only x -terms and constants (like 6x + 8 ) must be a perimeter.
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Question 13 of 36 Drag and drop Drag the expanded form into the box under each product. Not every tile is used.
(x + 4)(x - 4) (x - 4)^2 (x + 4)^2 Select… Select… Select…
Tap a tile, then tap a box to place it.
x^2 - 8x - 16 x^2 - 8x + 16 x^2 + 16 x^2 - 16 x^2 + 8x + 16
x² - 8x - 16 x² - 8x + 16 x² + 16 x² - 16 x² + 8x + 16 Hint Use the special products: (a + b)(a - b) = a^2 - b^2 and (a \pm b)^2 = a^2 \pm 2ab + b^2 .
Step-by-step solution (x + 4)(x - 4) : First x^2 , Outer -4x , Inner +4x , Last -16 . The middle terms cancel: x^2 - 16 (x - 4)^2 = (x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16 (x + 4)^2 = (x + 4)(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16 Unused tiles: x^2 + 16 (squaring each term of (x + 4)^2 separately) and x^2 - 8x - 16 (taking (-4)(-4) as -16 ).
Check: using x = 5 : (9)(1) = 9 and 25 - 16 = 9 ✓; (1)^2 = 1 and 25 - 40 + 16 = 1 ✓; (9)^2 = 81 and 25 + 40 + 16 = 81 ✓
Tip: a square of a binomial always ends in a positive constant, and a sum times a difference has no middle term.
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Type your answer in the box. Fractions like 3/4 are fine.
Hint Divide each term by 3x . Divide the coefficients and subtract the exponents.
Step-by-step solution \dfrac{6x^3}{3x} = 2x^{3 - 1} = 2x^2 .\dfrac{-9x^2}{3x} = -3x^{2 - 1} = -3x .\dfrac{3x}{3x} = 1 .Result: 2x^2 - 3x + 1 . So a = 2 , b = -3 , c = 1
Check: multiply back: 3x(2x^2 - 3x + 1) = 6x^3 - 9x^2 + 3x ✓
Tip: any non-zero term divided by itself is 1 , not 0 . Forgetting this is the most common mistake on these questions. Also keep each sign with its term: -9x^2 \div 3x is -3x , so b is negative.
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A small beverage company sells x cases of soda each week. Its weekly revenue, in dollars, is -x^2 + 60x , and its weekly cost, in dollars, is 12x + 150 .
Profit equals revenue minus cost. Which expression represents the weekly profit, in dollars?
Answer choices Hint Write Profit = (revenue) - (cost) with parentheses around the cost, then flip the sign of each cost term.
Step-by-step solution Profit = (-x^2 + 60x) - (12x + 150) . Distribute the minus sign: -(12x + 150) = -12x - 150 . Rewrite: -x^2 + 60x - 12x - 150 . x^2 -term: -x^2 (only one).x -terms: 60x - 12x = 48x .Constant: -150 . Weekly profit: -x^2 + 48x - 150
Check: let x = 10 cases. Revenue: -100 + 600 = 500 . Cost: 120 + 150 = 270 . Profit: 500 - 270 = 230 . Expression: -100 + 480 - 150 = 230 ✓
Tip: always put parentheses around the whole expression being subtracted. The cost includes the fixed \$150 , and that amount must be subtracted too.
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Question 16 of 36 Select all Which of these products simplify to 9x^2 - 25 ? Choose every correct answer.
Select all that apply.
Answer choices Hint 9x^2 - 25 = (3x)^2 - 5^2 , a difference of squares. Expand each choice, or test each one with a value such as x = 1 (the target value is 9 - 25 = -16 ).
Step-by-step solution Use (a + b)(a - b) = a^2 - b^2 with a = 3x and b = 5 : 9x^2 - 25 = (3x + 5)(3x - 5) .
(3x - 5)(3x + 5) = 9x^2 - 25 ✓(3x - 5)^2 = 9x^2 - 30x + 25 ✗ (a square, not a difference of squares)(5 + 3x)(3x - 5) : the order of terms inside a factor does not matter, 5 + 3x = 3x + 5 , so this is 9x^2 - 25 ✓(3x + 5)(5 - 3x) = 25 - 9x^2 ✗ (the opposite sign)(9x + 5)(x - 5) = 9x^2 - 45x + 5x - 25 = 9x^2 - 40x - 25 ✗-(5 - 3x)(5 + 3x) = -(25 - 9x^2) = 9x^2 - 25 ✓The equivalent expressions are (3x - 5)(3x + 5) , (5 + 3x)(3x - 5) and -(5 - 3x)(5 + 3x)
Check: at x = 1 the target is -16 . Choice 1: (-2)(8) = -16 ✓. Choice 3: (8)(-2) = -16 ✓. Choice 6: -(2)(8) = -16 ✓. Choice 4: (8)(2) = 16 ✗.
Tip: in a difference of squares, the term that comes first in the "minus" factor decides which square is positive: (a + b)(a - b) = a^2 - b^2 .
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Find the product (x + 3)(2x^2 - x + 4) and combine like terms. Which result is correct?
Answer choices Hint Multiply x by all three terms of the trinomial, then multiply 3 by all three terms. You should have six products before combining.
Step-by-step solution Distribute x : x(2x^2 - x + 4) = 2x^3 - x^2 + 4x . Distribute 3 : 3(2x^2 - x + 4) = 6x^2 - 3x + 12 . Add the two results: 2x^3 - x^2 + 4x + 6x^2 - 3x + 12 . x^2 -terms: -x^2 + 6x^2 = 5x^2 .x -terms: 4x - 3x = x .The product is 2x^3 + 5x^2 + x + 12
Check: let x = 1 . Original: (4)(2 - 1 + 4) = 4 \cdot 5 = 20 . Answer: 2 + 5 + 1 + 12 = 20 ✓
Tip: a binomial times a trinomial gives 2 \times 3 = 6 products. Count them before you combine; if you have fewer than six, a term was skipped.
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A student must simplify the quotient below, where x \ne 0 . Which answer is correct?
\frac{12x^4 - 8x^3 + 20x^2}{4x^2}
Answer choices Hint Split the fraction into three fractions, one for each term on top. For each, divide the coefficients and subtract the exponents.
Step-by-step solution Split: \dfrac{12x^4}{4x^2} - \dfrac{8x^3}{4x^2} + \dfrac{20x^2}{4x^2} . First: \dfrac{12}{4}x^{4-2} = 3x^2 . Second: \dfrac{8}{4}x^{3-2} = 2x , so the term is -2x . Third: \dfrac{20}{4}x^{2-2} = 5x^0 = 5 . The quotient is 3x^2 - 2x + 5
Check: multiply back: 4x^2(3x^2 - 2x + 5) = 12x^4 - 8x^3 + 20x^2 ✓. Or let x = 1 : \dfrac{12 - 8 + 20}{4} = \dfrac{24}{4} = 6 and 3 - 2 + 5 = 6 ✓
Tip: dividing by a monomial means dividing every term by it. Remember x^0 = 1 : when the exponents are equal, the variable disappears.
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Fencing will go all the way around a rectangular dog run that measures (x^2 + 2x) feet by (3x - 1) feet. Which polynomial gives the length of fencing needed (the perimeter ), in feet?
Answer choices Hint Perimeter is the distance around: add all four sides, or use P = 2l + 2w . Do not multiply length by width; that is area.
Step-by-step solution Perimeter of a rectangle: P = 2l + 2w . Substitute: P = 2(x^2 + 2x) + 2(3x - 1) . Distribute: 2(x^2 + 2x) = 2x^2 + 4x and 2(3x - 1) = 6x - 2 . Add: 2x^2 + 4x + 6x - 2 = 2x^2 + 10x - 2 . The perimeter is 2x^2 + 10x - 2
Check: let x = 3 . Length = 9 + 6 = 15 ft, width = 9 - 1 = 8 ft, perimeter = 2(15) + 2(8) = 46 ft. Expression: 2(9) + 10(3) - 2 = 18 + 30 - 2 = 46 ✓
Tip: perimeter adds lengths, so the highest power does not grow. Area multiplies lengths, so the powers add. If your "perimeter" has x^3 , you found an area.
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Dana simplified (6x^2 - 2x + 1) - (4x^2 + 3x - 5) . Her work is shown.
Step Work 1 6x^2 - 2x + 1 - 4x^2 - 3x + 5 2 (6x^2 - 4x^2) + (-2x - 3x) + (1 + 5) 3 2x^2 + x + 6
In which step did Dana make her first error?
Answer choices Hint Check each step on its own. Did the signs flip correctly in Step 1? Are the groups in Step 2 the same terms? Is each group simplified correctly in Step 3?
Step-by-step solution Step 1: subtracting the second polynomial flips every sign: -(4x^2 + 3x - 5) = -4x^2 - 3x + 5 . Dana wrote exactly that. Correct.Step 2: she grouped the like terms 6x^2 and -4x^2 , -2x and -3x , 1 and 5 , keeping every sign. Correct.Step 3: 6x^2 - 4x^2 = 2x^2 ✓ and 1 + 5 = 6 ✓, but -2x - 3x = -5x , not +x . Error. The first error is in Step 3 . The correct answer is 2x^2 - 5x + 6 .
Check: let x = 1 . Original: (6 - 2 + 1) - (4 + 3 - 5) = 5 - 2 = 3 . Correct answer: 2 - 5 + 6 = 3 ✓. Dana's answer: 2 + 1 + 6 = 9 ✗
Tip: two negative terms combine to a bigger negative: -2x - 3x = -5x . Plugging in x = 1 is a fast way to test any step.
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Question 21 of 36 Drop-down Subtract (2x^3 - x^2 + 6) from (5x^3 + 4x^2 - 3x) and simplify.
In the result, the coefficient of x^2 is Select… 5 3 -5 4 Select… -6 6 0
Choose an answer from each menu.
Hint "Subtract A from B" means B - A . The polynomial after the word from comes first.
Step-by-step solution "Subtract A from B " means B - A : (5x^3 + 4x^2 - 3x) - (2x^3 - x^2 + 6) . Flip every sign of the subtracted polynomial: 5x^3 + 4x^2 - 3x - 2x^3 + x^2 - 6 . x^3 -terms: 5x^3 - 2x^3 = 3x^3 .x^2 -terms: 4x^2 + x^2 = 5x^2 .x -term: -3x (only one).Constant: -6 (the first polynomial has no constant). The result is 3x^3 + 5x^2 - 3x - 6 : the coefficient of x^2 is 5 and the constant term is -6
Check: let x = 1 . B = 5 + 4 - 3 = 6 , A = 2 - 1 + 6 = 7 , B - A = -1 . Result: 3 + 5 - 3 - 6 = -1 ✓
Tip: read "subtract ... from ..." backwards when you write it. A missing term (like the constant in B ) counts as 0, so 0 - 6 = -6 .
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Type your answer in the box. Fractions like 3/4 are fine.
Hint Multiply 5xy by each term. Multiply coefficients, and add the exponents of x and of y separately.
Step-by-step solution 5xy \cdot 2x = 10x^{1+1}y = 10x^2y , so a = 10 .5xy \cdot (-3y) = -15xy^{1+1} = -15xy^2 , so b = -15 .5xy \cdot 1 = 5xy , so c = 5 .The product is 10x^2y - 15xy^2 + 5xy : a = 10 , b = -15 , c = 5
Check: let x = 1 , y = 2 . Original: 5(1)(2)(2 - 6 + 1) = 10(-3) = -30 . Answer: 10(1)(2) - 15(1)(4) + 5(1)(2) = 20 - 60 + 10 = -30 ✓
Tip: with two variables, treat each letter separately: x \cdot x = x^2 but x \cdot y stays xy . Do not forget the last term: 5xy \cdot 1 = 5xy , not 0 .
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Type your answer in the box. Fractions like 3/4 are fine.
Hint Use FOIL. Both binomials have a negative second term, so watch the signs of the Outer, Inner and Last products.
Step-by-step solution First: 2x \cdot 3x = 6x^2 .Outer: 2x \cdot (-2) = -4x .Inner: -7 \cdot 3x = -21x .Last: (-7)(-2) = +14 .Combine the middle terms: -4x - 21x = -25x . The product is 6x^2 - 25x + 14 : a = 6 , b = -25 , c = 14
Check: let x = 1 . Original: (2 - 7)(3 - 2) = (-5)(1) = -5 . Answer: 6 - 25 + 14 = -5 ✓
Tip: when both binomials have the form (\_ - \_) , the middle term is negative and the constant is positive. Use that pattern to catch sign slips.
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A rectangular banner has an area of (6x^3 + 15x^2 - 9x) square inches. Its width is 3x inches. Which expression represents the length of the banner, in inches?
Answer choices Hint Area = length \times width, so length = area \div width. Divide each term of the area by 3x .
Step-by-step solution Length = \dfrac{\text{area}}{\text{width}} = \dfrac{6x^3 + 15x^2 - 9x}{3x} . \dfrac{6x^3}{3x} = 2x^{3-1} = 2x^2 .\dfrac{15x^2}{3x} = 5x^{2-1} = 5x .\dfrac{-9x}{3x} = -3 .The length is 2x^2 + 5x - 3
Check: multiply back: 3x(2x^2 + 5x - 3) = 6x^3 + 15x^2 - 9x ✓. With x = 2 : area = 48 + 60 - 18 = 90 , width = 6 , length = 90 \div 6 = 15 , and 2(4) + 10 - 3 = 15 ✓
Tip: when a problem gives an area and one side, divide to find the other side. Always check by multiplying your answer by the side you were given.
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Question 25 of 36 Select all Select all expressions that are equivalent to (x - 2)(x^2 + 3x - 1) .
Select all that apply.
Answer choices Hint Multiply x by each term of the trinomial, then -2 by each term. You should get 6 products before combining.
Step-by-step solution x(x^2 + 3x - 1) = x^3 + 3x^2 - x .-2(x^2 + 3x - 1) = -2x^2 - 6x + 2 .Add: x^3 + 3x^2 - 2x^2 - x - 6x + 2 = x^3 + x^2 - 7x + 2 . Now test each choice:
x^3 + x^2 - 7x + 2 ✓x(x^2 + 3x - 1) - 2(x^2 + 3x - 1) ✓ (this is step 1 and 2 before expanding)(x^2 + 3x - 1)(x - 2) ✓ (the same factors in the other order)x^3 + 5x^2 + 5x - 2 ✗ (this is (x + 2)(x^2 + 3x - 1) )x^3 + 3x^2 - x - 2 ✗ (only x was distributed)The equivalent expressions are x^3 + x^2 - 7x + 2 , x(x^2 + 3x - 1) - 2(x^2 + 3x - 1) and (x^2 + 3x - 1)(x - 2)
Check: with x = 3 : (1)(9 + 9 - 1) = 17 and 27 + 9 - 21 + 2 = 17 ✓; the wrong choices give 27 + 45 + 15 - 2 = 85 and 27 + 27 - 3 - 2 = 49 .
Tip: the order of factors does not matter in multiplication, and an unsimplified form is still equivalent.
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Which expression is equivalent to (2x^2 + 5x - 3) \div (2x - 1) for x \ne \tfrac{1}{2} ?
Answer choices Hint Ask: what binomial times (2x - 1) gives 2x^2 + 5x - 3 ? Think about the first and last terms.
Step-by-step solution The quotient times 2x - 1 must give 2x^2 + 5x - 3 . First term: 2x \cdot (\;?\;) = 2x^2 , so the quotient starts with x . Last term: -1 \cdot (\;?\;) = -3 , so the quotient ends with +3 . Try x + 3 : (2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3 ✓ So \dfrac{(2x - 1)(x + 3)}{2x - 1} = x + 3 . The answer is x + 3
Check: with x = 2 : \dfrac{8 + 10 - 3}{4 - 1} = \dfrac{15}{3} = 5 , and 2 + 3 = 5 ✓
Tip: when the leading coefficient is not 1, use the first and last terms to guess the factor, then check the middle term by multiplying.
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The trinomial 3x^2 - 14x - 5 has (x - 5) as one of its factors. What is the result when 3x^2 - 14x - 5 is divided by x - 5 (with x \ne 5 )?
Answer choices Hint Factor the numerator. One of its factors should be (x - 5) , which cancels with the denominator.
Step-by-step solution Look for (x - 5)(\;?\;) = 3x^2 - 14x - 5 . The first term of the missing factor must be 3x (because x \cdot 3x = 3x^2 ). The last term must multiply with -5 to give -5 , so it is +1 . Test: (x - 5)(3x + 1) = 3x^2 + x - 15x - 5 = 3x^2 - 14x - 5 ✓ Cancel the common factor: \dfrac{(x - 5)(3x + 1)}{x - 5} = 3x + 1 . The quotient is 3x + 1
Check: let x = 2 . Original: \dfrac{12 - 28 - 5}{2 - 5} = \dfrac{-21}{-3} = 7 . Answer: 3(2) + 1 = 7 ✓
Tip: on multiple choice you can also multiply each answer by (x - 5) and see which one rebuilds the numerator. You may only cancel factors , never single terms.
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Marcus simplified 2(x + 3)^2 - (x^2 - 4) . His work is shown.
Step Work 1 2(x^2 + 6x + 9) - (x^2 - 4) 2 2x^2 + 12x + 18 - x^2 - 4 3 x^2 + 12x + 14
Which statement is true?
Answer choices Hint Check the square in Step 1 with (a + b)^2 = a^2 + 2ab + b^2 . Then check that the minus sign in Step 2 reached every term in (x^2 - 4) .
Step-by-step solution Step 1: (x + 3)^2 = x^2 + 2(3)x + 9 = x^2 + 6x + 9 . Correct.Step 2: 2(x^2 + 6x + 9) = 2x^2 + 12x + 18 ✓. But -(x^2 - 4) = -x^2 + 4 ; Marcus wrote -x^2 - 4 . Error. Step 3 combines Step 2 correctly, but it carries the error forward. Correct simplification: 2x^2 + 12x + 18 - x^2 + 4 = x^2 + 12x + 22 . So the true statement is: the first error is in Step 2, and the correct result is x^2 + 12x + 22 .
Check: let x = 1 . Original: 2(16) - (1 - 4) = 32 + 3 = 35 . Correct result: 1 + 12 + 22 = 35 ✓. Marcus's result: 1 + 12 + 14 = 27 ✗
Tip: "minus a negative" becomes a plus. Whenever a minus sign sits in front of parentheses, flip every sign inside, including the last one.
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The product (x + k)(x - 3) is equal to x^2 + 4x - 21 for every value of x .
What is the value of k ?
k =
Type your answer in the box. Fractions like 3/4 are fine.
Hint Multiply out (x + k)(x - 3) in terms of k . Then match the constant term (and check with the middle term).
Step-by-step solution Expand: (x + k)(x - 3) = x^2 - 3x + kx - 3k = x^2 + (k - 3)x - 3k . Match the constants: -3k = -21 , so k = \dfrac{-21}{-3} = 7 . Match the middle terms as a check: k - 3 = 4 , so k = 7 ✓ (both agree). The value is k = 7
Check: (x + 7)(x - 3) = x^2 - 3x + 7x - 21 = x^2 + 4x - 21 ✓
Tip: when two polynomials are equal for every x , the coefficients of matching powers must be equal. Use one coefficient to find the unknown and another to confirm it.
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Which expression is equivalent to \left(\tfrac{1}{2}x + 3\right)(4x - 2) ?
Answer choices Hint FOIL works the same with fractions: \tfrac{1}{2} \cdot 4 = 2 and \tfrac{1}{2} \cdot (-2) = -1 .
Step-by-step solution First: \tfrac{1}{2}x \cdot 4x = 2x^2 .Outer: \tfrac{1}{2}x \cdot (-2) = -x .Inner: 3 \cdot 4x = 12x .Last: 3 \cdot (-2) = -6 .Combine the middle terms: -x + 12x = 11x . The product is 2x^2 + 11x - 6
Check: let x = 2 . Original: (1 + 3)(8 - 2) = 4 \cdot 6 = 24 . Answer: 2(4) + 11(2) - 6 = 8 + 22 - 6 = 24 ✓
Tip: pick an even test value (like x = 2 ) when a problem has halves; the arithmetic stays in whole numbers and the check takes seconds.
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Type your answer in the box. Fractions like 3/4 are fine.
Hint There are 3 \times 3 = 9 products. Organize them by power of x (a table or box method helps). One coefficient may turn out to be 0.
Step-by-step solution Distribute x^2 : x^2(x^2 + x - 1) = x^4 + x^3 - x^2 . Distribute -2x : -2x(x^2 + x - 1) = -2x^3 - 2x^2 + 2x . Distribute 3 : 3(x^2 + x - 1) = 3x^2 + 3x - 3 . x^3 -terms: x^3 - 2x^3 = -x^3 , so b = -1 .x^2 -terms: -x^2 - 2x^2 + 3x^2 = 0 , so c = 0 .x -terms: 2x + 3x = 5x , so d = 5 .The product is x^4 - x^3 + 5x - 3 : b = -1 , c = 0 , d = 5
Check: let x = 2 . Original: (4 - 4 + 3)(4 + 2 - 1) = 3 \cdot 5 = 15 . Answer: 16 - 8 + 0 + 10 - 3 = 15 ✓
Tip: when like terms cancel completely, the coefficient is 0; the power is "missing" from the answer. Count your products (9 here) so none is skipped.
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Type your answer in the box. Fractions like 3/4 are fine.
Hint Uncovered area = (area of the whole yard) - (area of the shed). Find each area as a polynomial first.
Step-by-step solution Whole yard: (2x + 3)(x + 5) = 2x^2 + 10x + 3x + 15 = 2x^2 + 13x + 15 . Shed: x \cdot x = x^2 . Uncovered area: (2x^2 + 13x + 15) - x^2 = x^2 + 13x + 15 . So a = 1 , b = 13 , c = 15
Check: let x = 4 . Yard: 11 \times 9 = 99 sq ft. Shed: 16 sq ft. Uncovered: 99 - 16 = 83 . Expression: 16 + 52 + 15 = 83 ✓
Tip: "area left over" problems are always big area minus small area . The coefficient a = 1 is easy to miss: x^2 means 1x^2 .
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For some constant a , the product (ax + 5)(2x - 1) is equal to 6x^2 + 7x - 5 for every value of x . What is the value of a ?
Answer choices Hint Expand in terms of a . Which term of the product contains a alone with x^2 ?
Step-by-step solution Expand: (ax + 5)(2x - 1) = 2ax^2 - ax + 10x - 5 = 2ax^2 + (10 - a)x - 5 . Match the x^2 -terms: 2a = 6 , so a = 3 . Confirm with the x -terms: 10 - a = 10 - 3 = 7 ✓ (matches 7x ). The constants already agree: -5 = -5 . The value is a = 3
Check: (3x + 5)(2x - 1) = 6x^2 - 3x + 10x - 5 = 6x^2 + 7x - 5 ✓
Tip: use the term that gives the simplest equation (here the leading term), then confirm with a second coefficient to catch mistakes.
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Type your answer in the box. Fractions like 3/4 are fine.
Hint Path area = (area of the big outer square) − (area of the garden). How long is a side of the outer square?
Step-by-step solution The path adds 3 ft on each side, so the outer square's side is x + 3 + 3 = x + 6 feet. Outer area: (x + 6)^2 = x^2 + 12x + 36 (using (a + b)^2 = a^2 + 2ab + b^2 ). Garden area: x^2 . Path area: (x^2 + 12x + 36) - x^2 = 12x + 36 . So a = 12 and b = 36
Check: if x = 10 , the outer square is 16 \times 16 = 256 ft² and the garden is 100 ft², so the path is 156 ft². The expression gives 12(10) + 36 = 156 ✓
Tip: a border adds its width twice to each dimension (once on each side). Using x + 3 instead of x + 6 gives 6x + 9 , a common wrong answer.
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Question 35 of 36 Drop-down Lena multiplied (2x - 1)(x^2 + 4x - 3) . Her work is shown.
Step Work 1 2x(x^2 + 4x - 3) - 1(x^2 + 4x - 3) 2 2x^3 + 8x^2 - 6x - x^2 + 4x + 3 3 2x^3 + 7x^2 - 2x + 3
Lena's first error is in Select… Step 1 Step 2 Step 3 no step (the work is correct) Select… 2x³ + 7x² - 10x + 3 2x³ + 7x² - 2x + 3 2x³ + 9x² - 10x + 3 2x³ + 7x² - 10x - 3
Choose an answer from each menu.
Hint In Step 2, multiply -1 by each of x^2 , 4x and -3 yourself and compare with Lena's three terms.
Step-by-step solution Step 1: splitting (2x - 1) into 2x and -1 and distributing each is correct.Step 2: 2x(x^2 + 4x - 3) = 2x^3 + 8x^2 - 6x ✓. Then -1(x^2 + 4x - 3) = -x^2 - 4x + 3 . Lena wrote +4x instead of -4x . Error. Correct Step 2: 2x^3 + 8x^2 - 6x - x^2 - 4x + 3 . Combine: x^2 -terms 8x^2 - x^2 = 7x^2 ; x -terms -6x - 4x = -10x . Lena's first error is in Step 2 , and the correct product is 2x^3 + 7x^2 - 10x + 3
Check: let x = 1 . Original: (1)(1 + 4 - 3) = 2 . Correct product: 2 + 7 - 10 + 3 = 2 ✓. Lena's answer: 2 + 7 - 2 + 3 = 10 ✗
Tip: multiplying by -1 flips every sign in the parentheses. Test your final answer with x = 1 ; it takes ten seconds and catches most slips.
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A polynomial q(x) satisfies (x - 2) \cdot q(x) = x^3 - 7x + 6 for every value of x . Which expression is q(x) ?
Answer choices Hint q(x) = (x^3 - 7x + 6) \div (x - 2) . Remember that x^3 - 7x + 6 has no x^2 -term; write it as x^3 + 0x^2 - 7x + 6 before dividing.
Step-by-step solution q(x) = \dfrac{x^3 + 0x^2 - 7x + 6}{x - 2} (insert 0x^2 as a placeholder).Synthetic division with 2 (from x - 2 ), coefficients 1, 0, -7, 6 :bring down 1 ; 1 \cdot 2 = 2 , 0 + 2 = 2 ;2 \cdot 2 = 4 , -7 + 4 = -3 ;-3 \cdot 2 = -6 , 6 + (-6) = 0 (remainder 0 ✓). The quotient coefficients 1, 2, -3 give x^2 + 2x - 3 . So q(x) = x^2 + 2x - 3
Check: (x - 2)(x^2 + 2x - 3) = x^3 + 2x^2 - 3x - 2x^2 - 4x + 6 = x^3 - 7x + 6 ✓
Tip: a missing power needs a 0 placeholder in long or synthetic division. On multiple choice, multiplying each choice by (x - 2) is often just as fast.
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