SAT — mathFoundational5 min read

Simplifying Polynomial Expressions

Master adding, subtracting, and multiplying polynomials for the Digital SAT. Combine like terms and expand expressions with confidence.

Tutor AI Team2026-02-25T01:20:08.788Z

Polynomial expressions form the backbone of Advanced Math on the Digital SAT. Before you can factor, solve quadratics, or work with polynomial functions, you need to be able to simplify polynomials — adding, subtracting, and multiplying them fluently. This is a foundational skill that saves time and prevents errors on harder problems.

Core Concepts

What Is a Polynomial?

A polynomial is an expression with one or more terms, where each term has a coefficient and a variable raised to a non-negative integer power.

$3x^3 - 2x^2 + 5x - 7$

Terms: $3x^3$ , $-2x^2$ , $5x$ , $-7$
Degree: 3 (highest power)
Leading coefficient: 3

Like Terms

Like terms have the same variable(s) raised to the same power(s).

$5x^2$ and $-3x^2$ are like terms
$4x$ and $4x^2$ are NOT like terms

Adding Polynomials

Combine like terms:

$(3x^2 + 5x - 2) + (x^2 - 3x + 4)$ $= 4x^2 + 2x + 2$

Subtracting Polynomials

Distribute the negative sign, then combine:

$(5x^2 + 2x - 1) - (3x^2 - 4x + 6)$ $= 5x^2 + 2x - 1 - 3x^2 + 4x - 6$ $= 2x^2 + 6x - 7$

Multiplying Polynomials

Monomial × Polynomial: distribute the monomial: $3x(2x^2 - 5x + 1) = 6x^3 - 15x^2 + 3x$

Binomial × Binomial (FOIL): $(x + 3)(x - 7) = x^2 - 7x + 3x - 21 = x^2 - 4x - 21$

Binomial × Trinomial: distribute each term in the binomial: $(x + 2)(x^2 - 3x + 4)$ $= x(x^2 - 3x + 4) + 2(x^2 - 3x + 4)$ $= x^3 - 3x^2 + 4x + 2x^2 - 6x + 8$ $= x^3 - x^2 - 2x + 8$

Squaring a Binomial

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

Example: $(2x + 3)^2 = 4x^2 + 12x + 9$

Common error: $(x + 3)^2 \neq x^2 + 9$ . Don't forget the middle term!

Difference of Squares Pattern

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

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

Strategy Tips

Tip 1: Line Up Like Terms Vertically

For complex additions/subtractions, stack polynomials with like terms aligned.

Tip 2: Always Distribute the Negative

When subtracting, change every sign in the polynomial being subtracted.

Tip 3: Count Your Terms After Multiplying

Before combining, a binomial × binomial should produce 4 terms. A binomial × trinomial should produce 6 terms. Use this to catch errors.

Tip 4: Use the Special Product Formulas

Memorising $(a+b)^2$ , $(a-b)^2$ , and $(a+b)(a-b)$ speeds up SAT calculations.

Tip 5: Watch for SAT Shortcuts

Sometimes the SAT asks for the value of an expression like $(x+y)^2 - (x-y)^2$ . Using the identity: $= 4xy$ . Knowing these saves time.

Worked Example: Example 1

Problem

Simplify $(4x^2 - 3x + 7) + (2x^2 + 5x - 4)$

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

Solution

Worked Example: Example 2

Problem

Simplify $(6x^3 + x - 5) - (2x^3 - 3x^2 + x + 1)$

$= 6x^3 + x - 5 - 2x^3 + 3x^2 - x - 1$

$= 4x^3 + 3x^2 - 6$

Solution

Worked Example: Example 3

Problem

Expand $(3x - 2)(4x + 5)$

$= 12x^2 + 15x - 8x - 10 = 12x^2 + 7x - 10$

Solution

Worked Example: Example 4

Problem

Expand $(x + 4)^2 - (x - 4)^2$

Using the difference of squares: $= [(x+4) + (x-4)][(x+4) - (x-4)]$

$= [2x][8] = 16x$

Or expand directly: $(x^2 + 8x + 16) - (x^2 - 8x + 16) = 16x$

Solution

Worked Example: SAT-Style

Problem

If $f(x) = x^2 + 3x$ and $g(x) = 2x - 1$ , what is $f(x) \cdot g(x)$ ?

$= (x^2 + 3x)(2x - 1)$ $= 2x^3 - x^2 + 6x^2 - 3x$ $= 2x^3 + 5x^2 - 3x$

Solution

Practice Problems

Problem 1

Simplify $(3x^2 - 7x + 2) + (x^2 + 4x - 9)$ .

Problem 2

Simplify $(5x^3 + 2x) - (3x^3 - x^2 + 2x - 1)$ .

Problem 3

Expand $2x(x^2 - 4x + 3)$ .

Problem 4

Expand $(2x - 5)(3x + 1)$ .

Problem 5

Expand and simplify $(x + 6)^2$ .

Problem 6

Simplify $(x + 1)(x^2 - x + 1)$ .

Want to check your answers and get step-by-step solutions?

Common Mistakes

Forgetting the middle term in $(a+b)^2$ . The answer is $a^2 + 2ab + b^2$ , not $a^2 + b^2$ .
Sign errors when subtracting. Distribute the negative to EVERY term.
Missing terms when multiplying. Each term in one polynomial must multiply each term in the other.
Combining unlike terms. $x^2$ and $x$ are different — can't be combined.
Exponent errors. $x \cdot x^2 = x^3$ (add exponents), not $x^2$ .

Frequently Asked Questions

What's the degree of a polynomial?

The highest power of the variable. For $3x^4 - 2x + 1$ , the degree is 4.

Can I use FOIL for anything besides binomial × binomial?

FOIL is specifically for two binomials. For larger products, use the distributive property.

Do I need to know polynomial division for the SAT?

Rarely. Some harder questions may involve it, but most SAT polynomial questions focus on adding, subtracting, multiplying, and factoring.

What does "equivalent expression" mean?

Two expressions are equivalent if they produce the same value for every input. Simplifying or expanding gives an equivalent expression.

How does this connect to other SAT topics?

Polynomial simplification is foundational for factoring, solving quadratics, and working with polynomial functions.

Key Takeaways

✓
Add/subtract: combine like terms (same variable, same power).
✓
Multiply: distribute each term; use FOIL for binomial × binomial.
✓
Special products: $(a±b)^2 = a^2 ± 2ab + b^2$ and $(a+b)(a-b) = a^2 - b^2$ .
✓
Don't forget the middle term when squaring binomials.
✓
Distribute the negative when subtracting an entire polynomial.
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These skills are prerequisites for factoring, quadratics, and polynomial functions.

Tags:

#sat#math#advanced-math#polynomials

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Core Concepts

What Is a Polynomial?

Like Terms

Adding Polynomials

Subtracting Polynomials

Multiplying Polynomials

Squaring a Binomial

Difference of Squares Pattern

Strategy Tips

Tip 1: Line Up Like Terms Vertically

Tip 2: Always Distribute the Negative

Tip 3: Count Your Terms After Multiplying

Tip 4: Use the Special Product Formulas

Tip 5: Watch for SAT Shortcuts

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: Example 4

Worked Example: SAT-Style

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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