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Evaluating Polynomial Expressions

Find the value of polynomial expressions and formulas, including negatives, powers and real-world models.

Practice this skill 20 min Evaluate polynomials Leer en español
Key idea

What you'll learn

In this lesson you will evaluate polynomial expressions — expressions with powers such as x^2 and x^3 — for given values of the variables. You will learn to:

  • apply the order of operations when exponents are involved,
  • square and cube negative numbers correctly, and tell (-3)^2 apart from -3^2,
  • evaluate expressions with two variables, decimals and fractions,
  • use real-world polynomial formulas: the height of a thrown ball, revenue, the volume of a box,
  • work backwards to find a missing coefficient.

Why it matters on the GED: many formulas are polynomials. The test may give you a formula for height, profit or volume and ask for its value at one point, or ask you to compare two values. The wrong answer choices are carefully built from exponent and sign mistakes, so accuracy is everything.

Words to know

  • Polynomial — a sum of terms with whole-number exponents (-16t^2 + 64t + 5).
  • Base and exponent — in x^3, x is the base and 3 is the exponent: x^3 = x \cdot x \cdot x.
  • Squared — raised to the power 2 (5^2 = 25). Cubed — raised to the power 3 (2^3 = 8).
  • Coefficient — the number in front of a power; in 4x^2, the coefficient 4 multiplies x^2 (it is not squared).
  • Evaluate — substitute numbers for the variables and compute the value.
  • Function notation — p(x) names a polynomial; p(3) means "the value of p when x = 3".
  • Leading negative — the minus sign in -x^2; it means -(x^2), the opposite of x squared.
  • Even / odd power — an even power of a negative number is positive ((-2)^4 = 16); an odd power is negative ((-2)^3 = -8).

All Math words in the glossary

Exponents come before multiplication

The order of operations puts exponents before multiplication. So in 4x^2, you square x first and then multiply by 4.

4x^2 \text{ at } x = 3: \quad 4(3)^2 = 4 \cdot 9 = 36 \qquad \text{(not } (4 \cdot 3)^2 = 144\text{)}
Picture of 4x² at x = 3: four 3-by-3 squares, each with 3² = 9 small squares, for 4 · 9 = 36 in all. A red box shows the wrong method (4 · 3)² = 144, because only x is squared, not the coefficient.4x² at x = 3: 4 · (3)² = 4 · 9 = 363² = 93² = 93² = 93² = 94 squares of 9 = 36 ✓Wrong: (4 · 3)²= 12² = 144only x is squaredExponent first, then multiply by the coefficient.

Method:

  1. Substitute every variable with the number in parentheses.
  2. Work out all powers.
  3. Do all multiplication (coefficients times powers).
  4. Add and subtract from left to right.

Example: evaluate 2x^2 - 5x + 3 at x = 4.

StepWork
Substitute2(4)^2 - 5(4) + 3
Powers2 \cdot 16 - 5(4) + 3
Multiply32 - 20 + 3
Add/subtract15

Squaring is not doubling. x^2 means x \cdot x. At x = 3, x^2 = 9, not 6. And x^3 = x \cdot x \cdot x; at x = 2, x^3 = 8, not 6.

Higher powers grow quickly: at x = 3, x^2 = 9, x^3 = 27, x^4 = 81.

Negative numbers and powers

This is the biggest trap on the GED.

ExpressionMeaningValue
(-3)^2(-3)(-3)9
-3^2-(3 \cdot 3)-9
(-2)^3(-2)(-2)(-2)-8
-x^2 at x = -2-(-2)^2 = -(4)-4
x^2 at x = -5(-5)^225

Rules:

  • A negative number squared (or raised to any even power) is positive.
  • A negative number cubed (or raised to any odd power) is negative.
  • In -x^2, the exponent belongs to x only. Square x first, then take the opposite. So -x^2 is always zero or negative, whatever x is.
Bar chart of powers of −2: (−2)¹ = −2 and (−2)³ = −8 are negative (red bars below the axis); (−2)² = 4 and (−2)⁴ = 16 are positive (blue bars above). Even powers are positive and odd powers negative. Side boxes compare (−3)² = 9 with −3² = −9.Powers of −2: the sign alternates−2(−2)¹4(−2)²−8(−2)³16(−2)⁴even power → positive, odd power → negative(−3)² = (−3)(−3)= 9−3² = −(3 · 3)= −9

Example: evaluate -x^2 + 4x at x = -2.

  1. Substitute: -(-2)^2 + 4(-2).
  2. Power: (-2)^2 = 4, so the first term is -4.
  3. Multiply: 4(-2) = -8.
  4. Add: -4 + (-8) = -12.

Example: evaluate x^3 - 2x^2 + 5 at x = -2: (-2)^3 - 2(-2)^2 + 5 = -8 - 2(4) + 5 = -8 - 8 + 5 = -11.

Two variables, real formulas and working backwards

Two variables. Substitute each one in its own parentheses, then follow the same order.

a^2 - 2ab + b^2 at a = 5, b = -1: (5)^2 - 2(5)(-1) + (-1)^2 = 25 + 10 + 1 = 36.

Plug-in diagram for a² − 2ab + b² with a = 5 and b = −1: substitute to get (5)² − 2(5)(−1) + (−1)², work out powers and products to get 25 + 10 + 1, and add to get 36.a² − 2ab + b²substitute(5)² − 2(5)(−1) + (−1)²powers, then multiply25 + 10 + 1add36a = 5b = −1

Real-world formulas.

  • The height in feet of a ball thrown upward is often h = -16t^2 + vt + h_0, where t is seconds, v is the starting speed and h_0 the starting height.
  • Revenue can be modeled by R = -2p^2 + 120p, where p is the price.
  • The volume of an open box made by cutting x-inch squares from the corners of a sheet is length × width × height.
A rectangular sheet 10 inches by 8 inches with an x-by-x inch square cut from each of its four corners. Dashed lines show where the sides fold up to make an open box with height x, length 10 - 2x and width 8 - 2x.10 in8 inxbaseCut an x-by-x square from each corner, fold up the sides

For the sheet above, the box is 10 - 2x long (a square is cut from both ends), 8 - 2x wide, and x high, so V = x(10 - 2x)(8 - 2x).

Comparing two values. To find "how much more" one value is than another, evaluate the polynomial at both inputs and subtract.

Working backwards. If a polynomial has an unknown coefficient k and you know one value, substitute and solve. If p(x) = 2x^2 + kx - 5 and p(-2) = 7, then 2(4) + k(-2) - 5 = 7, so 3 - 2k = 7, and k = -2.

Formula

Rules to remember (not on the formula sheet)

x^2 = x \cdot x \qquad x^3 = x \cdot x \cdot x \qquad (-a)^2 = a^2 \qquad (-a)^3 = -a^3 \qquad -a^2 = -(a^2)
  • x is any number; a is any positive number.
  • Order of operations: parentheses → exponents → multiplication and division → addition and subtraction.

The exponent and order-of-operations rules are not on the GED formula sheet. Many polynomial formulas you will evaluate are on it — for example the area of a square A = s^2, the area of a circle A = \pi r^2, the volume of a sphere V = \tfrac{4}{3}\pi r^3, and the volume of a rectangular prism V = lwh. Word-problem formulas like the height of a ball are given in the question.

Worked example

Example 1 (easy): A quadratic expression

Evaluate 2x^2 - 3x + 1 when x = 4.

  1. Substitute: 2(4)^2 - 3(4) + 1.
  2. Power: (4)^2 = 16.
  3. Multiply: 2 \cdot 16 = 32 and 3 \cdot 4 = 12.
  4. Combine: 32 - 12 + 1 = 21.

The figure sums it up:

Plug-in steps for 2x² − 3x + 1 at x = 4: substitute to get 2(4)² − 3(4) + 1, square to get 2 · 16 − 3(4) + 1, multiply to get 32 − 12 + 1, then combine to get 21.2x² − 3x + 1substitute in ( )2(4)² − 3(4) + 1power: (4)² = 162 · 16 − 3(4) + 1multiply32 − 12 + 1add and subtract21x = 4

Answer: 21

Check: on the calculator: 2 × 4 x² − 3 × 4 + 1 enter → 21 ✓

Worked example

Example 2: A cubic with a negative value

Evaluate x^3 - 2x^2 + 5 when x = -2.

  1. Substitute in parentheses: (-2)^3 - 2(-2)^2 + 5.
  2. Powers: (-2)^3 = -8 (odd power, negative) and (-2)^2 = 4 (even power, positive).
  3. Multiply: 2 \cdot 4 = 8.
  4. Combine: -8 - 8 + 5 = -11.

The figure sums it up:

Plug-in steps for x³ − 2x² + 5 at x = −2: substitute to get (−2)³ − 2(−2)² + 5; powers give −8 − 2(4) + 5 because the odd power is negative and the even power positive; multiply to get −8 − 8 + 5; combine to get −11.x³ − 2x² + 5substitute in ( )(−2)³ − 2(−2)² + 5powers: odd −, even +−8 − 2(4) + 5multiply−8 − 8 + 5combine−11x = −2

Answer: -11

Check: (-2)(-2)(-2) = 4 \cdot (-2) = -8 ✓ and -8 - 8 = -16, -16 + 5 = -11 ✓

Worked example

Example 3 (word problem): Height of a ball

A ball is thrown upward. Its height in feet after t seconds is h = -16t^2 + 48t + 6. How high is the ball after 2 seconds?

  1. Substitute t = 2: h = -16(2)^2 + 48(2) + 6.
  2. Power: (2)^2 = 4.
  3. Multiply: -16 \cdot 4 = -64 and 48 \cdot 2 = 96.
  4. Combine: -64 + 96 + 6 = 38.

The figure sums it up:

Graph of the ball height h = −16t² + 48t + 6 for t from 0 to about 3.1 seconds. The curve starts at (0, 6), rises to a peak of 42 feet at t = 1.5, and comes down. The points (1, 38) and (2, 38) are at the same height, joined by a dashed line; after 2 seconds the ball is 38 feet high.h = −16t² + 48t + 60102030405000.511.522.533.5time t (seconds)feet(0, 6)(1, 38)(2, 38)after 2 s: 38 ft

Answer: 38 feet

Check: the ball starts at 6 ft (t = 0). After 1 second it is at -16 + 48 + 6 = 38 ft — the same height as at 2 seconds, which makes sense because it goes up and comes back down symmetrically ✓

Worked example

Example 4 (word problem): Handshakes

If n people each shake hands with every other person once, the number of handshakes is \dfrac{n^2 - n}{2}. How many handshakes happen at a meeting of 12 people?

  1. Substitute n = 12: \dfrac{(12)^2 - 12}{2}.
  2. Power: 12^2 = 144.
  3. Numerator: 144 - 12 = 132.
  4. Divide: 132 \div 2 = 66.

The figure sums it up:

Handshake diagrams: 3 people A, B, C joined by 3 lines (3 handshakes, and (9 − 3) ÷ 2 = 3), and 4 people A, B, C, D joined by 6 lines (6 handshakes, and (16 − 4) ÷ 2 = 6). For 12 people the formula gives (144 − 12) ÷ 2 = 66.Handshakes = (n² − n) ÷ 2ABCABCD3 people: 3 handshakes(9 − 3) ÷ 2 = 34 people: 6 handshakes(16 − 4) ÷ 2 = 612 people: (144 − 12) ÷ 2 = 66 handshakes

Answer: 66 handshakes

Check: with 3 people (A, B, C) there are 3 handshakes: AB, AC, BC. The formula gives \dfrac{9 - 3}{2} = 3 ✓

Worked example

Example 5 (GED-hard): Find the missing coefficient

The polynomial p(x) = 2x^2 + kx - 5 has the value p(-2) = 7. What is k?

  1. Substitute x = -2: 2(-2)^2 + k(-2) - 5 = 7.
  2. Power: (-2)^2 = 4, so 2 \cdot 4 = 8.
  3. Simplify: 8 - 2k - 5 = 7, so 3 - 2k = 7.
  4. Subtract 3: -2k = 4.
  5. Divide by -2: k = -2.

The figure sums it up:

Steps to find k in p(x) = 2x² + kx − 5 with p(−2) = 7: substitute to get 2(−2)² + k(−2) − 5 = 7, simplify to 8 − 2k − 5 = 7, then 3 − 2k = 7, subtract 3 to get −2k = 4, and divide by −2 to get k = −2.p(−2) = 7substitute x = −22(−2)² + k(−2) − 5 = 7power and multiply8 − 2k − 5 = 7combine 8 − 53 − 2k = 7subtract 3−2k = 4divide by −2k = −2x = −2

Answer: k = -2

Check: p(-2) = 2(4) + (-2)(-2) - 5 = 8 + 4 - 5 = 7 ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

The calculator follows the order of operations, so you can type a polynomial exactly as written — if you handle negatives correctly.

  • Square a number: press the x² key. 4 x² gives 16.
  • Other powers: use the ^ key. 2 ^ 3 gives 8. Press the right arrow to leave the exponent before typing more.
  • Negative inputs need parentheses. ( (−) 2 ) x² gives 4. Typing (−) 2 x² without parentheses gives -4, because the calculator squares 2 first, then takes the opposite — just like -2^2.
  • Example: -16t^2 + 48t + 6 at t = 2: (−) 1 6 × 2 x² + 4 8 × 2 + 6 enter → 38.

The calculator is especially useful when the inputs are decimals, like t = 1.5, or when the numbers are large.

When not to use it: small whole-number substitutions such as x^2 + 3x - 4 at x = 3 are quicker by hand, and doing them mentally builds the number sense you need on the rest of the test.

Common mistake

Common mistakes

  • Doubling instead of squaring. Wrong: x^2 at x = 3 is 6. Right: 3 \cdot 3 = 9.
  • Squaring the coefficient too. Wrong: -16t^2 at t = 1.5 is (-16 \cdot 1.5)^2 = 576. Right: square t only: -16(2.25) = -36.
  • Losing the sign of a negative input. Wrong: (-2)^2 = -4. Right: (-2)^2 = 4.
  • Ignoring a leading negative. Wrong: -x^2 at x = -2 is 4. Right: -(-2)^2 = -(4) = -4.
  • Negative times negative in a middle term. Wrong: -2ab at a = 5, b = -1 is -10. Right: -2(5)(-1) = +10.
  • Answering with one value when the question asks for a difference. Read "how much more" carefully: evaluate twice, then subtract.
Test strategy

GED test strategy

  • Write every substitution with parentheses on your erasable note board. For x^2 at x = -3, write (-3)^2 = 9, not -3^2 = -9.
  • Use the calculator for decimals, but type negatives inside parentheses.
  • Estimate to catch big errors. For -16t^2 + 64t + 5 at t = 1.5, the ball cannot be 677 feet high a second and a half after being thrown at a modest speed.
  • Look at the answer choices. If two choices differ only in sign, recheck your powers of negative numbers.
  • Work-backwards questions: you can also test each answer choice for the missing coefficient in the polynomial and see which one gives the stated value.
  • Units: height in feet, volume in cubic inches, money in dollars. Make sure the answer you pick has the right meaning.

Check yourself

Evaluate polynomialsNo calculator

What is the value of x^2 + 3x - 4 when x = 3?

Answer choices
Hint

Square 3 first (3 \cdot 3), then multiply 3 \cdot 3 for the middle term, then add and subtract.

Check yourself

Evaluate polynomialsNo calculatorFill in

Evaluate 2x^2 - 5 when x = -3.

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Square -3 first. A negative number times itself is positive. Then multiply by 2.

Summary

  • Substitute in parentheses, then: powers → multiply/divide → add/subtract.
  • x^2 means x \cdot x (not 2x); only the base is squared, not the coefficient.
  • A negative number to an even power is positive; to an odd power it is negative.
  • -x^2 means "the opposite of x^2", so -x^2 at x = -2 is -4.
  • For "how much more", evaluate at both values and subtract.
  • To find a missing coefficient, substitute the known values and solve.
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