Factoring

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Factoring: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Factoring means rewriting an expression as a product of simpler expressions, the reverse of multiplying, such as x² + 7x + 12 = (x + 3)(x + 4).

Algebra 1AlgebraA-SSE.A.2

What is Factoring?

Multiplying builds a product from factors; factoring goes backward and finds the factors hiding inside a sum. Numbers do it too: \(12 = 3 \cdot 4\).

For expressions, factoring often starts by pulling out a greatest common factor, then looks for patterns such as a trinomial \(x^2 + bx + c\) that splits into \((x + p)(x + q)\) with \(pq = c\) and \(p + q = b\). A correct answer always multiplies back to the original expression, so you can check it.

How to spot it

  • A question says factor completely, write as a product or find the factors.
  • A quadratic equation must be set to zero and solved by splitting it into pieces.
  • An expression has a common number or variable in every term.
  • A rectangle’s area is given as a polynomial and you need its length and width.

The method

  1. Pull out the greatest common factor of all terms, if there is one other than 1.
  2. For \(x^2 + bx + c\), list pairs of integers with product c and choose the pair whose sum is b.
  3. Write the two binomial factors using that pair.
  4. Multiply the factors to check that you get the original expression.
x²4x3x12x4x3x² + 4x + 3x + 12= x² + 7x + 12= (x + 3)(x + 4)3 · 4 = 12 and 3 + 4 = 7
The area x² + 7x + 12 is a rectangle with sides (x + 3) and (x + 4).

Factoring: solved example

Problem

A rectangular patio has area \(x^2 + 9x + 20\) square feet. Factor the expression to find possible side lengths.

Solution

Find two integers with product 20 and sum 9. The pairs of 20 are 1 and 20, 2 and 10, 4 and 5; only 4 and 5 add to 9.

So \(x^2 + 9x + 20 = \)\((x + 4)(x + 5)\). Check: \(x^2 + 5x + 4x + 20 = x^2 + 9x + 20\).

Try it

Question 1

Factor \(3x^2 + 6x\).

Show the answer

The common factor is \(3x\): \(3x(x + 2)\).

Question 2

Factor \(x^2 – 5x – 14\).

Show the answer

Product \(-14\), sum \(-5\): the numbers are \(-7\) and 2. So \((x – 7)(x + 2)\).

Factoring vs. FOIL Method

FOIL and this skill are opposites. FOIL turns a product of binomials into a trinomial; this skill turns the trinomial back into the product. If you are stuck factoring, multiply a guess with FOIL and see whether the original comes back.

FOIL Method

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