SAT Polynomials & Factoring: Complete Guide + Practice

Polynomial manipulation appears in SAT Advanced Math, especially through:

• equivalent expressions;
• nonlinear equations;
• nonlinear functions.

The SAT often rewards seeing structure quickly.

Combine like terms

[ 3x^2+5x-2+x^2-7x+9 ]

[ 4x^2-2x+7 ]

Working confidently with exponents like these is easier once you've reviewed exponents and radicals.

Distribute

[ 2x(x+5)=2x^2+10x ]

Difference of squares

[ a^2-b^2=(a-b)(a+b) ]

Example:

[ x^2-49=(x-7)(x+7) ]

Factoring trinomials

[ x^2+7x+12 ]

Find numbers that multiply to 12 and add to 7:

3 and 4.

So:

[ (x+3)(x+4) ]

Factor out the GCF first

[ 6x^2+18x ]

[ 6x(x+3) ]

Students often miss easy factoring because they jump directly into a more complicated method.

Roots from factors

If:

[ (x-5)(x+2)=0 ]

then:

[ x=5 ]

or:

[ x=-2 ]

Remember:

factor (x-r) corresponds to root (r).

If solving for x afterward feels shaky, our linear equations guide covers the fundamentals these roots build on.

Equivalent expressions

The SAT may ask which expression is equivalent to:

[ (x+4)^2-16 ]

Expand:

[ x^2+8x+16-16=x^2+8x ]

Factor:

[ x(x+8) ]

All three forms can be useful for different questions.

Use structure before expanding

Suppose:

[ (x-3)^2=25 ]

You do not need to expand.

[ x-3=\pm5 ]

So:

[ x=8,\ -2 ]

That same structure-first approach carries over to the quadratic equations guide, especially for questions solved by factoring or completing the square.

Original practice

1

Factor:

[ x^2-81 ]

2

Factor:

[ x^2+9x+20 ]

3

Factor completely:

[ 8x^2-24x ]

4

If:

[ (x+6)(x-1)=0 ]

what are the roots?

5

Simplify:

[ (2x+3)(x-4) ]

Answers

1. ((x-9)(x+9))
2. ((x+4)(x+5))
3. (8x(x-3))
4. -6 and 1
5. (2x^2-5x-12)

Common traps

• sign errors between roots and factors;
• forgetting a greatest common factor;
• expanding when factoring would be faster;
• combining unlike terms.

For more Advanced Math skill guides, see our SAT Math guides.

Official sources

• College Board Advanced Math
  https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced
• College Board Student Question Bank Math skill descriptions
  https://satsuite.collegeboard.org/practice/student-question-bank/math

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