Polynomials and Factoring

Polynomials are the foundation of quadratics and many other topics. Factoring is one of the key tools for solving them, so the more comfortable you become with factoring, the easier quadratics will feel.

Click here for some tips on polynomials and factoring

Polynomial questions in Algebra 1 cover adding, subtracting, and multiplying polynomials, and factoring in three main forms. Factoring also comes up inside quadratic solving, so these skills build.

Here are some tips to keep in mind:

• Adding/subtracting early: combine like terms (same variable, same exponent)

• Multiplying (FOIL): First, Outer, Inner, Last for two binomials

• GCF (greatest common factor) factoring: always check this first — pull out the greatest common factor

• Trinomial factoring (x² + bx + c): find two numbers that multiply to c and add to b

• Difference of perfect squares: a² − b² = (a + b)(a − b).

  
  

✓ Always pull out the GCF before doing anything else. A problem like 3x² − 12 looks hard until you factor out 3: 3(x² − 4) = 3(x + 2)(x − 2). Exams frequently test whether you remember this first step.

Worked example

Problem: (x + 3)(x − 5)

Goal: use the FOIL (first, outer, inner, last)

1. First = x^2

2. Outer = -5x

3. Inner = 3x

4. Last = -15

5. Combine these to get x^2-2x-15

If you found this scratchpad helpful, get our Chrome extension to use it with almost any algebra problem!

Try it yourself! Below is a real Regents exam problem from January 2024

Problem: Factor completely: x² + 9x + 20. Then find the values of x that make the expression equal to zero.

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Click here to see the answer

STEP-BY-STEP SOLUTION

1. Factor: (x + 4)(x + 5) = 0

2. Set each factor to zero: x + 4 = 0 → x = −4   or   x + 5 = 0 → x = −5

✗ COMMON TRAP

Forgetting that (x+4) does not mean that x=4.

✓ THE FIX

Always remember that you need to get x alone for each factor if the question asks what values make the expression equal zero.

For more practice with polynomials and factoring, check out these resources:

https://momentofmath.com/mom.html? (look under "quadratic equations")