Polynomial Multiplication Worksheets

Multiply binomials with FOIL, from simple products to squaring a binomial. Grades 9–11.

Ctrl+P to print worksheet Ctrl+R to randomize

What is polynomial multiplication?

Multiplying polynomials is the distributive property used more than once: every term in the first polynomial multiplies every term in the second, and then like terms are combined. These worksheets focus on the most common case, a binomial times a binomial, which always produces at most three terms: an x2x^{2} term, an xx term and a constant.

FOIL is a memory aid for the four products: First terms, Outer terms, Inner terms and Last terms. The outer and inner products are both xx terms, so they combine into the middle term of the answer.

  • Easy — (x+a)(x+b)(x + a)(x + b): both binomials start with xx.
  • Medium — (ax+b)(cx+d)(ax + b)(cx + d) with leading coefficients from 2 to 5.
  • Hard — squaring a binomial, (ax+b)2(ax + b)^{2}.

Worked examples

Easy: (x+4)(x−7)(x + 4)(x - 7)

First x⋅x=x2x \cdot x = x^{2}, Outer x⋅(−7)=−7xx \cdot (-7) = -7x, Inner 4⋅x=4x4 \cdot x = 4x, Last 4⋅(−7)=−284 \cdot (-7) = -28. Combine the middle terms: −7x+4x=−3x-7x + 4x = -3x.

(x+4)(x−7)=x2−3x−28(x + 4)(x - 7) = x^{2} - 3x - 28

Medium: (2x−3)(3x+5)(2x - 3)(3x + 5)

First 6x26x^{2}, Outer 10x10x, Inner −9x-9x, Last −15-15.

(2x−3)(3x+5)=6x2+10x−9x−15=6x2+x−15(2x - 3)(3x + 5) = 6x^{2} + 10x - 9x - 15 = 6x^{2} + x - 15

Hard: (3x−2)2(3x - 2)^{2}

Squaring means multiplying the binomial by itself: (3x−2)(3x−2)(3x - 2)(3x - 2).

9x2−6x−6x+4=9x2−12x+49x^{2} - 6x - 6x + 4 = 9x^{2} - 12x + 4

This follows the pattern (a+b)2=a2+2ab+b2(a + b)^{2} = a^{2} + 2ab + b^{2} with a=3xa = 3x and b=−2b = -2: the middle term is 2(3x)(−2)=−12x2(3x)(-2) = -12x.

Check your answers

Substitute a small number for xx on both sides. For the medium example with x=2x = 2: (4−3)(6+5)=11(4 - 3)(6 + 5) = 11 and 6(4)+2−15=116(4) + 2 - 15 = 11. If the two values differ, there is a mistake.

Common mistakes

  • Dropping the middle term when squaring. (3x−2)2(3x - 2)^{2} is not 9x2+49x^{2} + 4. The outer and inner products do not cancel.
  • Squaring only the variable. (3x)2=9x2(3x)^{2} = 9x^{2}, not 3x23x^{2}.
  • Sign errors in the last term. (−3)(5)=−15(-3)(5) = -15: a negative times a positive is negative.
  • Combining unlike terms. 6x26x^{2} and xx are not like terms, so 6x2+x6x^{2} + x cannot be simplified further.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. (x−3)(x+8)=(x - 3)(x + 8) =
  2. (x+9)(x−6)=(x + 9)(x - 6) =
  3. (x−9)(x+2)=(x - 9)(x + 2) =

Medium

  1. (5x−7)(4x+1)=(5x - 7)(4x + 1) =
  2. (2x−6)(2x+3)=(2x - 6)(2x + 3) =
  3. (4x−6)(5x+5)=(4x - 6)(5x + 5) =

Hard

  1. (x−1)2=(x - 1)^{2} =
  2. (x−4)2=(x - 4)^{2} =
  3. (4x+5)2=(4x + 5)^{2} =
Show answers
  1. x² + 5x - 24
  2. x² + 3x - 54
  3. x² - 7x - 18
  4. 20x² - 23x - 7
  5. 4x² - 6x - 18
  6. 20x² - 10x - 30
  7. x² - 2x + 1
  8. x² - 8x + 16
  9. 16x² + 40x + 25

Tips for teachers and parents

  • A 2-by-2 area grid (the "box method") shows all four products at once and helps students who lose track of terms with FOIL.
  • Have students check one problem per row by substituting x=1x = 1 or x=2x = 2; it takes seconds and builds the habit.
  • The Hard level prepares students for recognising perfect-square trinomials when they move on to factoring.

Related topics