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Factoring Polynomials

Factoring is the process of rewriting an expression as a product of simpler expressions. It is the reverse of expanding brackets. Mastering factoring is essential for solving quadratic equations, simplifying fractions, and much of advanced algebra.

In this lesson

  1. What factoring means
  2. Step 1: Always check for GCF first
  3. Difference of squares
  4. Factoring trinomials (ax² + bx + c)
  5. Factoring by grouping

1 What Factoring Means

Factoring an expression means rewriting it as a multiplication. Just as 12 = 3 × 4, the polynomial x² + 5x + 6 = (x + 2)(x + 3). The factors (x + 2) and (x + 3) multiply together to give the original expression.

Why factor? Because factored form makes it easy to find zeros (where the expression equals zero), simplify fractions, and solve equations. If (x + 2)(x + 3) = 0, then x = −2 or x = −3 — immediate from factored form, but not obvious from x² + 5x + 6 = 0.

Factoring vs Expanding

Expanding: (x + 2)(x + 3) → x² + 5x + 6. Factoring: x² + 5x + 6 → (x + 2)(x + 3). They are inverse operations. FOIL (First, Outer, Inner, Last) is used to expand; factoring reverses this process.

2 Step 1: Always Check for GCF First

Before trying any other method, check whether all terms share a common factor. Factor it out first — this simplifies everything that follows.

GCF Factoring
Factor: 6x³ + 9x² − 3x
1Find the GCF of all terms: GCF(6, 9, 3) = 3; GCF(x³, x², x) = x; so GCF = 3x
2Factor out 3x: 3x(2x² + 3x − 1)
3Check by expanding: 3x(2x²) + 3x(3x) + 3x(−1) = 6x³ + 9x² − 3x ✓
Answer: 3x(2x² + 3x − 1)

3 Difference of Squares

The pattern a² − b² = (a + b)(a − b) is one of the most useful factoring identities. It applies whenever you have two perfect squares being subtracted.

Difference of Squares
Factor: x² − 25
1Recognize a² − b²: x² − 25 = x² − 5²
2Apply the pattern: (x + 5)(x − 5)
3Check: (x + 5)(x − 5) = x² − 5x + 5x − 25 = x² − 25 ✓
Answer: (x + 5)(x − 5)
Trickier Difference of Squares
Factor: 4x² − 49
1Recognize: 4x² = (2x)² and 49 = 7²
2Apply pattern: (2x + 7)(2x − 7)
3Check: (2x)² − 7² = 4x² − 49 ✓
Answer: (2x + 7)(2x − 7)
Sum of squares doesn't factor

a² + b² does NOT factor over the real numbers. x² + 25 cannot be factored using real numbers. Only the DIFFERENCE of squares factors this way.

4 Factoring Trinomials

For x² + bx + c (leading coefficient 1): find two numbers that multiply to c and add to b. These numbers become the constants in the factors.

Simple Trinomial
Factor: x² + 7x + 12
1Find two numbers that multiply to 12 AND add to 7
2Pairs that multiply to 12: (1,12), (2,6), (3,4)
3Which pair adds to 7? 3 + 4 = 7 ✓
4Write the factors: (x + 3)(x + 4)
5Check: x² + 4x + 3x + 12 = x² + 7x + 12 ✓
Answer: (x + 3)(x + 4)
Trinomial with Negative Terms
Factor: x² − 5x + 6
1Need two numbers that multiply to +6 and add to −5
2Both must be negative (multiply to positive, add to negative)
3Pairs: (−1,−6) add to −7; (−2,−3) add to −5 ✓
4Factors: (x − 2)(x − 3)
Answer: (x − 2)(x − 3)

5 Factoring by Grouping

For four-term polynomials, group terms in pairs, factor each pair, then factor out the common binomial.

Factoring by Grouping
Factor: 2x³ + 4x² + 3x + 6
1Group: (2x³ + 4x²) + (3x + 6)
2Factor each group: 2x²(x + 2) + 3(x + 2)
3Factor out the common binomial (x + 2): (x + 2)(2x² + 3)
Answer: (x + 2)(2x² + 3)

Try the Quadratic Formula Calculator

Once a polynomial is factored, use the quadratic formula calculator to verify the roots.

Open Calculator →

Practice Problems

Factor: x² + 9x + 20
Find two numbers multiplying to 20 and adding to 9: 4 and 5. Answer: (x + 4)(x + 5)
Factor: x² − 16
Difference of squares: x² − 4² = (x + 4)(x − 4)
Factor: 2x² + 8x
GCF = 2x. Answer: 2x(x + 4)
Factor: x² − 3x − 18
Need numbers multiplying to −18 and adding to −3: −6 and +3. Answer: (x − 6)(x + 3)