Factoring Polynomials Worksheets

Factor quadratic expressions, from simple trinomials to special forms and leading coefficients. Grades 9–11.

Ctrl+P to print worksheet Ctrl+R to randomize

What is factoring?

Factoring is multiplication in reverse. Instead of expanding (x+5)(x−3)(x + 5)(x - 3) into x2+2x−15x^{2} + 2x - 15, you start from the expanded form and find the factors that multiply to give it. Factoring is the key step in solving quadratic equations and in simplifying algebraic fractions.

Every expression on these worksheets is a quadratic that factors over the integers. The three levels cover the main patterns:

  • Easy — x2+bx+cx^{2} + bx + c: find two integers whose product is cc and whose sum is bb.
  • Medium — special forms: the difference of squares a2−b2=(a+b)(a−b)a^{2} - b^{2} = (a + b)(a - b) and the perfect-square trinomial a2+2ab+b2=(a+b)2a^{2} + 2ab + b^{2} = (a + b)^{2}.
  • Hard — a leading coefficient of 2, 3 or 4, as in ax2+bx+cax^{2} + bx + c, factored with the a⋅ca \cdot c (grouping) method.

Worked examples

Easy: x2+2x−15x^{2} + 2x - 15

Look for two integers with product −15-15 and sum 22. The pair 55 and −3-3 works.

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

Medium: x2−49x^{2} - 49 and x2−10x+25x^{2} - 10x + 25

x2−49x^{2} - 49 is a difference of squares, x2−72x^{2} - 7^{2}:

x2−49=(x+7)(x−7)x^{2} - 49 = (x + 7)(x - 7)

In x2−10x+25x^{2} - 10x + 25, the last term is 525^{2} and the middle term is 2⋅5⋅x2 \cdot 5 \cdot x, so it is a perfect square:

x2−10x+25=(x−5)2x^{2} - 10x + 25 = (x - 5)^{2}

Hard: 3x2+10x+83x^{2} + 10x + 8

Multiply a⋅c=3⋅8=24a \cdot c = 3 \cdot 8 = 24 and find two numbers with product 24 and sum 10: 66 and 44. Split the middle term and factor by grouping:

3x2+6x+4x+8=3x(x+2)+4(x+2)=(3x+4)(x+2)3x^{2} + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)

Check your answers

Multiply the factors back out. For the hard example, (3x+4)(x+2)=3x2+6x+4x+8=3x2+10x+8(3x + 4)(x + 2) = 3x^{2} + 6x + 4x + 8 = 3x^{2} + 10x + 8, which matches the original expression.

Common mistakes

  • Mixing up the signs. If cc is positive, both numbers have the same sign as bb. If cc is negative, the numbers have opposite signs.
  • Factoring a sum of squares. x2+49x^{2} + 49 does not factor over the integers; only a difference of squares does.
  • Leaving a common factor inside. Always take out the greatest common factor first: 2x2+10x+12=2(x2+5x+6)=2(x+2)(x+3)2x^{2} + 10x + 12 = 2(x^{2} + 5x + 6) = 2(x + 2)(x + 3). Writing (2x+4)(x+3)(2x + 4)(x + 3) gives the same product but is not completely factored.
  • Not checking. Multiplying the factors back together takes a few seconds and catches sign errors immediately.

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. x2−5x−6=x^{2} - 5x - 6 =
  2. x2+x−2=x^{2} + x - 2 =
  3. x2−4=x^{2} - 4 =

Medium

  1. x2−81=x^{2} - 81 =
  2. x2+4x+4=x^{2} + 4x + 4 =
  3. x2−12x+36=x^{2} - 12x + 36 =

Hard

  1. 3x2+11x−20=3x^{2} + 11x - 20 =
  2. 3x2−5x−12=3x^{2} - 5x - 12 =
  3. 4x2+29x+30=4x^{2} + 29x + 30 =
Show answers
  1. (x - 6)(x + 1)
  2. (x + 2)(x - 1)
  3. (x - 2)(x + 2)
  4. (x + 9)(x - 9)
  5. (x + 2)²
  6. (x - 6)²
  7. (3x - 4)(x + 5)
  8. (3x + 4)(x - 3)
  9. (4x + 5)(x + 6)

Tips for teachers and parents

  • Students who are slow on Easy usually need more practice with factor pairs of numbers up to 50; a quick list of factor pairs helps.
  • Pair this topic with Polynomial Multiplication: expanding and factoring the same expressions shows that they are inverse operations.
  • On Hard, ask students to write the a⋅ca \cdot c product and the chosen pair next to each problem so you can see where an error started.

Related topics