Factor polynomials

1. Find the greatest common factor of the numbers: 10 = 5 \cdot 2 and 15 = 5 \cdot 3, so the number part is 5.
2. Find the shared variable part: the terms have x^2 and x. The smallest power is x.
3. GCF = 5x.
4. Divide each term by the GCF: 10x^2 \div 5x = 2x and -15x \div 5x = -3.
5. Write the factored form: 5x(2x - 3).

Answer: 5x(2x - 3)

Check: 5x \cdot 2x - 5x \cdot 3 = 10x^2 - 15x ✓

Tip: "Complete" factorization means nothing inside the parentheses can be factored out again. If the terms inside still share a number or a variable, keep going.

1. We need two numbers that multiply to 15 and add to 8.
2. List the positive factor pairs of 15: 1 \cdot 15 (sum 16) and 3 \cdot 5 (sum 8) ✓.
3. Both numbers are positive because the constant (+15) and the middle term (+8x) are both positive.
4. So x^2 + 8x + 15 = (x + 3)(x + 5).

Answer: (x + 3)(x + 5)

Check: (x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15 ✓

Tip: A pair must pass both tests: the right product and the right sum. Checking only one test is the most common way to pick a wrong answer.

1. Numbers: the greatest common factor of 8 and 20 is 4 (since 8 = 4 \cdot 2 and 20 = 4 \cdot 5).
2. Variables: both terms contain at least x^2 (x^3 and x^2). The GCF uses the lowest power, so it is x^2.
3. GCF = 4x^2.
4. Divide each term by 4x^2: \dfrac{8x^3}{4x^2} = 2x and \dfrac{20x^2}{4x^2} = 5.
5. Write the product: 8x^3 + 20x^2 = 4x^2(2x + 5).

Answer: 4x^2(2x + 5)

Check: 4x^2 \cdot 2x = 8x^3 and 4x^2 \cdot 5 = 20x^2, so 4x^2(2x + 5) = 8x^3 + 20x^2 ✓. Inside the parentheses, 2x and 5 share no common factor, so the factoring is complete.

Tip: After factoring out a GCF, look inside the parentheses. If the terms there still share a factor, you did not take out the greatest one.

1. For x^2 + bx + c, find two numbers that multiply to c and add to b. Here c = 20 and b = -9.
2. The product is positive, so both numbers have the same sign. The sum is negative, so both are negative.
3. List negative pairs that multiply to 20: -1 and -20 (sum -21); -2 and -10 (sum -12); -4 and -5 (sum -9).
4. The pair -4 and -5 works.
5. So x^2 - 9x + 20 = (x - 4)(x - 5).

Answer: (x - 4)(x - 5)

Check: FOIL: x \cdot x = x^2; x(-5) + (-4)x = -9x; (-4)(-5) = 20. Total x^2 - 9x + 20 ✓.

Tip: Read the signs first. Positive c with negative b means both factors have a minus sign. That alone eliminates most wrong choices.

1. Look for two numbers that multiply to -30 and add to +1 (the coefficient of x).
2. A negative product means the numbers have opposite signs. The positive one must be larger because the sum is positive.
3. Try pairs: 6 and -5 give 6 \cdot (-5) = -30 and 6 + (-5) = 1 ✓.
4. So x^2 + x - 30 = (x + 6)(x - 5).
5. The factors are x + 6 and x - 5. Only x - 5 appears among the choices.

Answer: x - 5

Check: (x + 6)(x - 5) = x^2 - 5x + 6x - 30 = x^2 + x - 30 ✓. You can also test x = 5: 25 + 5 - 30 = 0. A value that makes the polynomial zero tells you x - 5 is a factor.

Tip: To test a choice like x - k, plug in x = k. If the polynomial equals 0, then x - k is a factor.

1. Area = length × width, so the area expression is the product of the two side expressions.
2. Factor x^2 + 3x - 28: we need two numbers that multiply to -28 and add to +3.
3. The product is negative, so the signs are opposite; the sum is positive, so the bigger number is positive.
4. Try pairs: 7 \cdot (-4) = -28 and 7 + (-4) = 3 ✓.
5. So x^2 + 3x - 28 = (x + 7)(x - 4).
6. The length is x + 7, so the width is the other factor.

Answer: x - 4 meters

Check: Let x = 10. Area = 100 + 30 - 28 = 102. Length = 17, width = 6, and 17 \cdot 6 = 102 ✓

Tip: Whenever a problem gives an area (a product) and one side, the missing side is the "other factor." Factoring is division in disguise.

1. Group: (x^3 - 5x^2) + (3x - 15).
2. Factor the first group: x^3 - 5x^2 = x^2(x - 5).
3. Factor the second group: 3x - 15 = 3(x - 5).
4. Now both parts share (x - 5): x^2(x - 5) + 3(x - 5).
5. Pull out the shared binomial: (x - 5)(x^2 + 3).

Answer: (x - 5)(x^2 + 3)

Check: (x - 5)(x^2 + 3) = x^3 + 3x - 5x^2 - 15 = x^3 - 5x^2 + 3x - 15 ✓

Tip: After you factor each group, the parentheses must match exactly. If they don't, try changing the sign you pull out of the second group.

1. Both terms are perfect squares: 25x^2 = (5x)^2 and 64 = 8^2.
2. There is a minus sign between them and no middle term, so this is a difference of squares.
3. Use a^2 - b^2 = (a - b)(a + b) with a = 5x and b = 8.
4. 25x^2 - 64 = (5x - 8)(5x + 8).

Answer: (5x - 8)(5x + 8)

Check: (5x - 8)(5x + 8) = 25x^2 + 40x - 40x - 64 = 25x^2 - 64 ✓. The middle terms cancel, as they always do for a difference of squares.

Tip: Take the square root of both the coefficient and the variable part: \sqrt{25x^2} = 5x, not 25x. Then write one factor with minus and one with plus.

1. Write x^2 - 11x + 28 = (x - 4)(x + n).
2. The constant: (-4) \cdot n = 28, so n = 28 \div (-4) = -7.
3. Check the sum: -4 + (-7) = -11 ✓, which matches the middle coefficient.
4. The other factor is (x - 7).

Answer: x - 7

Check: (x - 4)(x - 7) = x^2 - 7x - 4x + 28 = x^2 - 11x + 28 ✓.

Tip: When you know one factor, divide the constant by that factor's number to get the other number. Then confirm with the sum. Do not subtract the numbers.

1. Group in pairs: (x^3 + 4x^2) + (5x + 20).
2. Factor each pair:
   • x^3 + 4x^2 = x^2(x + 4)
   • 5x + 20 = 5(x + 4)
3. Now the expression is x^2(x + 4) + 5(x + 4). Both parts share the binomial (x + 4).
4. Factor it out: (x + 4)(x^2 + 5).

Answer: (x^2 + 5)(x + 4)

Check: (x^2 + 5)(x + 4) = x^3 + 4x^2 + 5x + 20 ✓.

Tip: In grouping, the binomials left in the parentheses after step 2 must match exactly. If they do not match, check your signs or try a different grouping. Note that x^2(x + 4) + 5(x + 4) is still a sum, not a product, until you finish step 4.

1. A perfect-square trinomial has the form (x - a)^2 = x^2 - 2ax + a^2.
2. Match the middle terms: -2a = -12, so a = 6.
3. The constant must be a^2 = 6^2 = 36.
4. So c = 36 and x^2 - 12x + 36 = (x - 6)^2.

Answer: c = 36

Check: (x - 6)^2 = x^2 - 6x - 6x + 36 = x^2 - 12x + 36 ✓.

Tip: Shortcut: take half of the middle coefficient, then square it. Half of -12 is -6, and (-6)^2 = 36. The constant in a perfect square is always positive.

1. Step 1 states the correct rule for x^2 + bx + c: product = c = -15, sum = b = -2 ✓.
2. Step 2: 3 \cdot (-5) = -15 ✓ and 3 + (-5) = -2 ✓. The numbers are right.
3. Step 3: the numbers 3 and -5 go into the factors with their own signs: (x + 3)(x - 5). Dana wrote (x - 3)(x + 5), flipping both signs.
4. Multiply her answer: (x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15. That has +2x, not -2x.

Answer: The first error is in Step 3; the correct factorization is (x + 3)(x - 5).

Check: (x + 3)(x - 5) = x^2 - 5x + 3x - 15 = x^2 - 2x - 15 ✓.

Tip: The number 3 goes into the factor as (x + 3), and -5 as (x - 5). Always multiply back to check, because sign slips are easy to miss.

1. Use the ac method: a \cdot c = 6 \cdot (-20) = -120.
2. Find two numbers that multiply to -120 and add to 7: 15 and -8 (since 15 \cdot (-8) = -120 and 15 + (-8) = 7).
3. Split the middle term: 6x^2 + 15x - 8x - 20.
4. Group: 3x(2x + 5) - 4(2x + 5).
5. Pull out (2x + 5): (3x - 4)(2x + 5).

Answer: (3x - 4)(2x + 5)

Check: (3x - 4)(2x + 5) = 6x^2 + 15x - 8x - 20 = 6x^2 + 7x - 20 ✓

Tip: Every choice here starts with 6x^2 and ends with -20. Only the middle term separates right from wrong, so always FOIL all four products.

1. GCF: every coefficient is divisible by 5 and every term has at least one x. GCF = 5x.
2. 5x^3 - 20x^2 - 60x = 5x(x^2 - 4x - 12).
3. Factor x^2 - 4x - 12: two numbers that multiply to -12 and add to -4 are -6 and 2.
4. x^2 - 4x - 12 = (x - 6)(x + 2).
5. Complete factorization: 5x(x - 6)(x + 2).

Answer: 5x(x - 6)(x + 2)

Check: (x - 6)(x + 2) = x^2 - 4x - 12; then 5x \cdot (x^2 - 4x - 12) = 5x^3 - 20x^2 - 60x ✓.

Tip: "Factored completely" is a two-step routine: GCF first, then ask if anything inside can still be factored. Stopping after the GCF is the most common trap.

1. ac-method: a \cdot c = 3 \cdot (-8) = -24.
2. Find two numbers that multiply to -24 and add to 10: 12 and -2.
3. Split the middle term: 3x^2 + 12x - 2x - 8.
4. Group: (3x^2 + 12x) + (-2x - 8) = 3x(x + 4) - 2(x + 4).
5. Factor out (x + 4): (3x - 2)(x + 4).

Answer: (3x - 2)(x + 4)

Check: FOIL: 3x \cdot x = 3x^2; outer 3x \cdot 4 = 12x; inner -2 \cdot x = -2x; last -2 \cdot 4 = -8. Middle: 12x - 2x = 10x ✓.

Tip: With a \ne 1, the position of each number matters. Always FOIL your answer: the outer and inner products must add to the middle term.

1. For a rectangle, \text{Area} = \text{length} \times \text{width}, so length = \text{Area} \div \text{width}.
2. Factor 2x^2 + 11x + 12. ac-method: 2 \cdot 12 = 24; two numbers that multiply to 24 and add to 11 are 8 and 3.
3. Split: 2x^2 + 8x + 3x + 12 = 2x(x + 4) + 3(x + 4) = (2x + 3)(x + 4).
4. The width is (x + 4), so the length is the other factor: 2x + 3.

Answer: 2x + 3 yards

Check: (2x + 3)(x + 4) = 2x^2 + 8x + 3x + 12 = 2x^2 + 11x + 12 ✓. With x = 10: width 14 yd, length 23 yd, area 322; and 2(100) + 110 + 12 = 322 ✓.

Tip: A missing side of a rectangle comes from dividing (factoring), never from subtracting the known side from the area.

1. Group: (2x^3 - 6x^2) + (-5x + 15).
2. First pair: 2x^3 - 6x^2 = 2x^2(x - 3).
3. Second pair: factor out -5 so the binomial matches: -5x + 15 = -5(x - 3).
4. Now: 2x^2(x - 3) - 5(x - 3).
5. Factor out (x - 3): (x - 3)(2x^2 - 5).

Answer: (x - 3)(2x^2 - 5)

Check: (x - 3)(2x^2 - 5) = 2x^3 - 5x - 6x^2 + 15 = 2x^3 - 6x^2 - 5x + 15 ✓.

Tip: If the second pair starts with a minus sign, factor out the negative. -5(x - 3) matches the first binomial; 5(-x + 3) does not look like it does.

1. Step 1: both terms are divisible by 2: 18x^2 \div 2 = 9x^2 and 50 \div 2 = 25. ✓
2. Step 2: 9x^2 = (3x)^2 and 25 = 5^2, so 9x^2 - 25 = (3x - 5)(3x + 5). ✓
3. Step 3: Ramon dropped the GCF of 2. The complete factorization is 2(3x - 5)(3x + 5).
4. Test his answer: (3x - 5)(3x + 5) = 9x^2 - 25, which is only half of 18x^2 - 50.

Answer: Step 3 is wrong; it should be 2(3x - 5)(3x + 5).

Check: 2(3x - 5)(3x + 5) = 2(9x^2 - 25) = 18x^2 - 50 ✓.

Tip: When you factor in stages, carry every factor you pulled out into the final line. A quick multiply-back catches a lost GCF immediately.

1. GCF: 3x^4 - 48 = 3(x^4 - 16).
2. x^4 - 16 = (x^2)^2 - 4^2 is a difference of squares: (x^2 - 4)(x^2 + 4).
3. x^2 - 4 = x^2 - 2^2 is again a difference of squares: (x - 2)(x + 2).
4. x^2 + 4 is a sum of squares, so it does not factor.
5. Complete factorization: 3(x^2 + 4)(x - 2)(x + 2).

Answer: 3(x^2 + 4)(x - 2)(x + 2)

Check: (x - 2)(x + 2) = x^2 - 4; (x^2 - 4)(x^2 + 4) = x^4 - 16; 3(x^4 - 16) = 3x^4 - 48 ✓

Tip: A difference of squares can hide inside another one. After each step, look at every factor and ask, "Is this a square minus a square?"

1. Group: (x^3 - 5x^2) + (-4x + 20).
2. Factor each pair: x^2(x - 5) - 4(x - 5). (Factor -4 from the second pair so the binomials match.)
3. Pull out (x - 5): (x - 5)(x^2 - 4).
4. x^2 - 4 is a difference of squares: (x - 2)(x + 2).
5. Complete factorization: (x - 5)(x - 2)(x + 2).

Answer: (x - 5)(x - 2)(x + 2)

Check: (x - 2)(x + 2) = x^2 - 4; (x - 5)(x^2 - 4) = x^3 - 4x - 5x^2 + 20 = x^3 - 5x^2 - 4x + 20 ✓. Also, x = 2 gives 8 - 20 - 8 + 20 = 0 ✓.

Tip: GED-hard factoring often chains two techniques. After grouping, always look at the new factors again: x^2 - 4, x^2 - 9, x^2 - 16 are signals to keep going.

1. Factor 2x^2 + 5x - 3 (ac-method): 2 \cdot (-3) = -6; numbers that multiply to -6 and add to 5 are 6 and -1.
2. Split: 2x^2 + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3).
3. Factor 4x^2 - 1 (difference of squares): (2x)^2 - 1^2 = (2x - 1)(2x + 1).
4. Factors of the first: 2x - 1 and x + 3. Factors of the second: 2x - 1 and 2x + 1.
5. The shared factor is 2x - 1.

Answer: 2x - 1

Check: (2x - 1)(x + 3) = 2x^2 + 6x - x - 3 = 2x^2 + 5x - 3 ✓ and (2x - 1)(2x + 1) = 4x^2 - 1 ✓. Both equal 0 at x = \tfrac{1}{2}.

Tip: A common factor of two polynomials must make both equal zero at the same x. Test a choice by finding the x that makes it zero and plugging that into each polynomial.

1. Jada: (2x + 3)^2 = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9 ✓. Correct.
2. Marco: (x + 3)^2 = x^2 + 6x + 9, which has a middle term 6x. x^2 + 9 is a sum of squares and does not factor. ✗
3. Leo: (x - 5)(x + 5) = x^2 - 25, a difference of squares. But x^2 - 10x + 25 is the perfect square (x - 5)^2. ✗
4. Only Jada's factorization is correct.

Answer: Only Jada is correct.

Check: (x - 5)^2 = x^2 - 10x + 25 ✓, confirming Leo should have written (x - 5)^2.

Tip: Squared binomials always produce a middle term; (a - b)(a + b) never does. Use the middle term to decide which pattern you have.