Polynomials and Factoring

Factor polynomials, perform polynomial arithmetic, understand the relationship between factors and zeros, and use the Remainder Theorem.

Written and reviewed by the Study Mondo Education TeamLast updated April 18, 2026

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Polynomials and Factoring on the SAT

What Is a Polynomial?

A polynomial is an expression with variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents.

$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

Degree: The highest power of the variable
Leading coefficient: The coefficient of the highest-degree term
Constant term: The term with no variable ( $a_0$ )

Types of Polynomials

Degree	Name	Example
0	Constant	$5$
1	Linear	$3x + 2$
2	Quadratic	$x^2 - 4x + 1$
3	Cubic	$2x^3 + x - 7$
4	Quartic	$x^4 - 3x^2 + 1$

Factoring Techniques

1. Greatest Common Factor (GCF)

$6x^3 + 9x^2 = 3x^2(2x + 3)$

2. Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$ Example: $x^2 - 25 = (x+5)(x-5)$

3. Perfect Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$ Example: $x^2 + 10x + 25 = (x+5)^2$

4. Trinomial Factoring ( $x^2 + bx + c$ )

Find two numbers that multiply to $c$ and add to $b$ . Example: $x^2 + 7x + 12 = (x+3)(x+4)$

5. Trinomial Factoring ( $ax^2 + bx + c$ , $a \neq 1$ )

Use the AC method or trial and error. Example: $2x^2 + 7x + 3 = (2x+1)(x+3)$

6. Sum and Difference of Cubes

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

7. Factor by Grouping

$x^3 + 3x^2 + 2x + 6 = x^2(x+3) + 2(x+3) = (x^2+2)(x+3)$

Polynomial Operations

Addition/Subtraction

Combine like terms (same variable, same exponent).

Multiplication

Use FOIL for binomials, or distribute each term. $(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$

Division

Polynomial long division or synthetic division (for dividing by $x - c$ ).

Remainder Theorem

When polynomial $P(x)$ is divided by $(x - c)$ , the remainder is $P(c)$ .

Example: If $P(x) = x^3 - 2x + 1$ , the remainder when divided by $(x - 3)$ is: $P(3) = 27 - 6 + 1 = 22$

Factor Theorem

$(x - c)$ is a factor of $P(x)$ if and only if $P(c) = 0$ .

Example: Is $(x - 2)$ a factor of $x^3 - 4x^2 + x + 6$ ? $P(2) = 8 - 16 + 2 + 6 = 0$ → Yes!

SAT Question Types

Type 1: Factor a Polynomial

"Factor $x^2 - 5x - 6$ " → $(x-6)(x+1)$

Type 2: Find Zeros from Factored Form

"If $f(x) = (x-2)(x+3)(x-5)$ , what are the zeros?" → $x = 2, -3, 5$

Type 3: Polynomial Division

"What is the remainder when $x^3 + 2x - 5$ is divided by $x - 1$ ?" $P(1) = 1 + 2 - 5 = -2$ (use the Remainder Theorem!)

Type 4: Equivalent Expressions

"Which expression is equivalent to $(x+2)^3$ ?" $= x^3 + 3(x^2)(2) + 3(x)(4) + 8 = x^3 + 6x^2 + 12x + 8$

Common SAT Mistakes

Sign errors when factoring — double-check by FOILing your answer
Forgetting the GCF before trying other methods
Confusing $(x+3)^2$ with $x^2 + 9$ — it's $x^2 + 6x + 9$ !
Not using the Remainder Theorem — much faster than long division
Dropping terms when subtracting polynomials — distribute the negative sign

📚 Practice Problems

1Problem 1easy

❓ Question:

Expand: $(x + 6)(x - 2)$

💡 Show Solution

Solution:

Use FOIL (First, Outer, Inner, Last):

F: $x \cdot x = x^2$ O: $x \cdot (-2) = -2x$ I: $6 \cdot x = 6x$ L: $6 \cdot (-2) = -12$

Combine: $x^2 - 2x + 6x - 12$ $= x^2 + 4x - 12$

Answer: $x^2 + 4x - 12$

SAT Tip: Don't forget to combine like terms after FOIL-ing!

2Problem 2medium

❓ Question:

Factor completely: $x^2 - 49$

💡 Show Solution

Solution:

Recognize difference of squares pattern: $a^2 - b^2 = (a+b)(a-b)$

Here: $x^2 - 49 = x^2 - 7^2$

Factor: $= (x + 7)(x - 7)$

Answer: $(x + 7)(x - 7)$

Check: $(x+7)(x-7) = x^2 - 7x + 7x - 49 = x^2 - 49$ ✓

SAT Tip: Difference of squares is common! Memorize $a^2 - b^2 = (a+b)(a-b)$

3Problem 3hard

❓ Question:

Which is equivalent to $(2x - 3)^2$ ?

A) $4x^2 - 9$ B) $4x^2 + 9$ C) $4x^2 - 12x + 9$ D) $4x^2 + 12x + 9$

💡 Show Solution

Solution:

Use square of a difference pattern: $(a - b)^2 = a^2 - 2ab + b^2$

Here: $a = 2x$ , $b = 3$

$(2x)^2 = 4x^2$ $-2(2x)(3) = -12x$ $3^2 = 9$

Result: $4x^2 - 12x + 9$

Answer: C

Common trap: A is $(2x+3)(2x-3)$ (difference of squares, not square!)

SAT Tip: $(a-b)^2$ has THREE terms, not two! Don't forget the middle term.

4Problem 4easy

❓ Question:

Factor completely: $3x^2 - 12$

💡 Show Solution

Step 1: Factor out the GCF first: $3x^2 - 12 = 3(x^2 - 4)$

Step 2: Recognize the difference of squares: $3(x^2 - 4) = 3(x + 2)(x - 2)$

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

Key: Always look for a GCF first!

5Problem 5easy

❓ Question:

Factor completely: $3x^2 - 12$

💡 Show Solution

Step 1: Factor out the GCF first: $3x^2 - 12 = 3(x^2 - 4)$

Step 2: Recognize the difference of squares: $3(x^2 - 4) = 3(x + 2)(x - 2)$

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

Key: Always look for a GCF first!

6Problem 6medium

❓ Question:

Factor: $2x^2 + 7x + 3$

💡 Show Solution

AC Method: $a \cdot c = 2 \cdot 3 = 6$ . Find two numbers that multiply to 6 and add to 7: 6 and 1.

Rewrite middle term: $2x^2 + 6x + x + 3$

Factor by grouping: $2x(x + 3) + 1(x + 3)$ $(2x + 1)(x + 3)$

Check: $(2x+1)(x+3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ ✓

Answer: $(2x+1)(x+3)$

7Problem 7medium

❓ Question:

Factor: $2x^2 + 7x + 3$

💡 Show Solution

AC Method: $a \cdot c = 2 \cdot 3 = 6$ . Find two numbers that multiply to 6 and add to 7: 6 and 1.

Rewrite middle term: $2x^2 + 6x + x + 3$

Factor by grouping: $2x(x + 3) + 1(x + 3)$ $(2x + 1)(x + 3)$

Check: $(2x+1)(x+3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ ✓

Answer: $(2x+1)(x+3)$

8Problem 8medium

❓ Question:

What is the remainder when $P(x) = 2x^3 - x^2 + 3x - 7$ is divided by $(x - 2)$ ?

💡 Show Solution

Use the Remainder Theorem: The remainder when $P(x)$ is divided by $(x - c)$ is $P(c)$ .

Here $c = 2$ : $P(2) = 2(2)^3 - (2)^2 + 3(2) - 7$ $= 2(8) - 4 + 6 - 7$ $= 16 - 4 + 6 - 7$ $= 11$

Answer: The remainder is 11.

SAT Tip: The Remainder Theorem saves enormous time compared to polynomial long division!

9Problem 9medium

❓ Question:

What is the remainder when $P(x) = 2x^3 - x^2 + 3x - 7$ is divided by $(x - 2)$ ?

💡 Show Solution

Use the Remainder Theorem: The remainder when $P(x)$ is divided by $(x - c)$ is $P(c)$ .

Here $c = 2$ : $P(2) = 2(2)^3 - (2)^2 + 3(2) - 7$ $= 2(8) - 4 + 6 - 7$ $= 16 - 4 + 6 - 7$ $= 11$

Answer: The remainder is 11.

SAT Tip: The Remainder Theorem saves enormous time compared to polynomial long division!

10Problem 10hard

❓ Question:

If $f(x) = x^3 - 6x^2 + 11x - 6$ and $f(1) = 0$ , factor $f(x)$ completely.

💡 Show Solution

Step 1: Since $f(1) = 0$ , by the Factor Theorem, $(x - 1)$ is a factor.

Step 2: Divide $x^3 - 6x^2 + 11x - 6$ by $(x - 1)$ using synthetic division:

$1 | 1 \quad -6 \quad 11 \quad -6$ $\quad | \quad 1 \quad -5 \quad 6$ $\quad 1 \quad -5 \quad 6 \quad 0$

Quotient: $x^2 - 5x + 6$

Step 3: Factor the quadratic: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Answer: $f(x) = (x-1)(x-2)(x-3)$

The zeros are $x = 1, 2, 3$ .

11Problem 11hard

❓ Question:

If $f(x) = x^3 - 6x^2 + 11x - 6$ and $f(1) = 0$ , factor $f(x)$ completely.

💡 Show Solution

Step 1: Since $f(1) = 0$ , by the Factor Theorem, $(x - 1)$ is a factor.

Step 2: Divide $x^3 - 6x^2 + 11x - 6$ by $(x - 1)$ using synthetic division:

$1 | 1 \quad -6 \quad 11 \quad -6$ $\quad | \quad 1 \quad -5 \quad 6$ $\quad 1 \quad -5 \quad 6 \quad 0$

Quotient: $x^2 - 5x + 6$

Step 3: Factor the quadratic: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Answer: $f(x) = (x-1)(x-2)(x-3)$

The zeros are $x = 1, 2, 3$ .

12Problem 12expert

❓ Question:

Which polynomial has zeros at $x = -1$ , $x = 2$ , and $x = 4$ , and passes through the point $(0, -16)$ ?

💡 Show Solution

Step 1: Write the general form using the zeros: $f(x) = a(x + 1)(x - 2)(x - 4)$

Step 2: Use the point $(0, -16)$ to find $a$ : $f(0) = a(0 + 1)(0 - 2)(0 - 4) = a(1)(-2)(-4) = 8a$ $-16 = 8a$ $a = -2$

Step 3: Write the final polynomial: $f(x) = -2(x+1)(x-2)(x-4)$

Check: $f(0) = -2(1)(-2)(-4) = -2(8) = -16$ ✓

Answer: $f(x) = -2(x+1)(x-2)(x-4)$

Expanded: $f(x) = -2x^3 + 10x^2 - 4x - 16$

13Problem 13expert

❓ Question:

Which polynomial has zeros at $x = -1$ , $x = 2$ , and $x = 4$ , and passes through the point $(0, -16)$ ?

💡 Show Solution

Step 1: Write the general form using the zeros: $f(x) = a(x + 1)(x - 2)(x - 4)$

Step 2: Use the point $(0, -16)$ to find $a$ : $f(0) = a(0 + 1)(0 - 2)(0 - 4) = a(1)(-2)(-4) = 8a$ $-16 = 8a$ $a = -2$

Step 3: Write the final polynomial: $f(x) = -2(x+1)(x-2)(x-4)$

Check: $f(0) = -2(1)(-2)(-4) = -2(8) = -16$ ✓

Answer: $f(x) = -2(x+1)(x-2)(x-4)$

Expanded: $f(x) = -2x^3 + 10x^2 - 4x - 16$

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❓ Frequently Asked Questions

What is Polynomials and Factoring?▾

Factor polynomials, perform polynomial arithmetic, understand the relationship between factors and zeros, and use the Remainder Theorem.

How can I study Polynomials and Factoring effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 13 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

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What course covers Polynomials and Factoring?▾

Polynomials and Factoring is part of the SAT Prep course on Study Mondo, specifically in the Advanced Math section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Polynomials and Factoring?▾

Yes, this page includes 13 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Polynomials and Factoring

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Polynomials and Factoring on the SAT

What Is a Polynomial?

Types of Polynomials

Factoring Techniques

1. Greatest Common Factor (GCF)

2. Difference of Squares

3. Perfect Square Trinomials

4. Trinomial Factoring (x2+bx+cx^2 + bx + cx2+bx+c)

5. Trinomial Factoring (ax2+bx+cax^2 + bx + cax2+bx+c, a≠1a \neq 1a=1)

6. Sum and Difference of Cubes

7. Factor by Grouping

Polynomial Operations

Addition/Subtraction

Multiplication

Division

Remainder Theorem

Factor Theorem

SAT Question Types

Type 1: Factor a Polynomial

Type 2: Find Zeros from Factored Form

Type 3: Polynomial Division

Type 4: Equivalent Expressions

Common SAT Mistakes

📚 Practice Problems

1Problem 1easy

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2Problem 2medium

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3Problem 3hard

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4Problem 4easy

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5Problem 5easy

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6Problem 6medium

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7Problem 7medium

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8Problem 8medium

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9Problem 9medium

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10Problem 10hard

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11Problem 11hard

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12Problem 12expert

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13Problem 13expert

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Practice Lab

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📌 Related Topics in Advanced Math

❓ Frequently Asked Questions

4. Trinomial Factoring ( $x^2 + bx + c$ )

5. Trinomial Factoring ( $ax^2 + bx + c$ , $a \neq 1$ )