SAT Math skill page

SAT Polynomials Practice

Use structure in expressions so factoring and expanding feel predictable instead of messy.

Practice polynomials in Dolphin SAT Practice questions

12-18 min practice time3 questions on pageAdvanced Math

Practice time12-18 min

Question bank3 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Combine like terms in polynomial expressions
Expand products of binomials
Factor simple quadratics
Recognize equivalent polynomial forms

Dolphin question bank

Practice your skill

Question 1 of 3Medium

Which of the following is equivalent to the sum of 3x^4 + 2x^3 and 4x^4 + 7x^3?

$16 x^{14}$
$7 x^{8} + 9 x^{6}$
$12 x^{4} + 14 x^{3}$
$7 x^{4} + 9 x^{3}$

Show answer and solution

Correct answer $D .7 x^{4} + 9 x^{3}$

Solution walkthrough

Adding the two expressions yields $3 x^{4} + 2 x^{3} + 4 x^{4} + 7 x^{3}$ .
Because the pair of terms $3 x^{4}$ and $4 x^{4}$ and the pair of terms $2 x^{3}$ and $7 x^{3}$ each contain the same variable raised to the same power, they are like terms and can be combined as $7 x^{4}$ and $9 x^{3}$ , respectively.
The sum of the given expressions therefore simplifies to $7 x^{4} + 9 x^{3}$ .

Question 2 of 3Medium

The sum of -2x^2 + x + 31 and 3x^2 + 7x - 8 can be written in the form ax^2 + bx + c, where a, b, and c are constants. What is the value of a + b + c?

Show answer and solution

Correct answerC. 32

Solution walkthrough

The correct answer is 32. The sum of the given expressions is $(- 2 x^{2} + x + 31) + (3 x^{2} + 7 x - 8)$ .
Combining like terms yields $x^{2} + 8 x + 23$ . Based on the form of the given equation, $a = 1$ , $b = 8$ , and $c = 23$ .
Therefore, $a + b + c = 32$ . Alternate approach: Because $a + b + c$ is the value of $a x^{2} + b x + c$ when $x = 1$ , it is possible to first make that substitution into each polynomial before adding them.
When $x = 1$ , the first polynomial is equal to $- 2 + 1 + 31 = 30$ and the second polynomial is equal to $3 + 7 - 8 = 2$ . The sum of 30 and 2 is 32.

Question 3 of 3Easy

Which of the following is equivalent to the expression (2x^3 + 3x)(

x^{3}

- 2x)?

$x^{3} + 5 x$
$3 x^{3} + x$
$2 x^{6} - x^{4} - 6 x^{2}$
$3 x^{6} - x^{4} - 6 x^{2}$

Show answer and solution

Correct answer $C .2 x^{6} - x^{4} - 6 x^{2}$

Solution walkthrough

Using the distributive property to multiply the terms in the parentheses yields $(2 x^{3} \cdot x^{3}) + (2 x^{3} \cdot - 2 x) + (3 x \cdot x^{3}) + (3 x \cdot - 2 x)$ , which is equivalent to $2 x^{6} - 4 x^{4} + 3 x^{4} - 6 x^{2}$ .
Combining like terms results in $2 x^{6} - x^{4} - 6 x^{2}$ .

Why the other choices miss

Choices A and D are incorrect and may result from conceptual errors when multiplying the terms in the given expression. Choice B is incorrect and may result from adding, instead of multiplying,

(2 x^{3} + 3 x)

and

(x^{3} - 2 x)

Avoid these traps

Common mistakes on this skill

Combining terms with different powers

Only like terms with the same variable part and exponent can be combined by adding coefficients.

Distributing a negative sign incompletely

A negative factor changes every term inside the parentheses.

Study plan

How to practice this skill in Dolphin

Arrange terms by descending exponent and identify like terms.
Use distribution or factoring one stage at a time, preserving every sign.
Expand the result or test multiple values to check the rewrite.

Practice polynomials in Dolphin SAT

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FAQ

Questions about SAT Polynomials Practice

What are like terms?

They have identical variable factors raised to identical powers; only their coefficients may differ.

How do I check a polynomial factorization?

Multiply the factors back together and confirm that the original polynomial is recovered.