3.3 Factoring & Rational Expressions

GCF Factoring

Factoring reverses multiplication: it rewrites a polynomial as a product of simpler factors. The FIRST step in every factoring problem is to pull out the greatest common factor (GCF) — the largest monomial that divides every term.

Worked example: Factor 12x³ + 18x².

• GCF of 12 and 18 is 6; the lowest power of x common to both is x².
• So the GCF is 6x²: 12x³ + 18x² = 6x²(2x + 3).
• Check by distributing: 6x²·2x + 6x²·3 = 12x³ + 18x². ✓

Factoring Trinomials: x² + bx + c

When the leading coefficient is 1, find two numbers that multiply to c and add to b. Those numbers become the constants in the two binomial factors.

Worked example: Factor x² + 7x + 12.

• Need two numbers with product 12 and sum 7 → 3 and 4.
• Factored form: (x + 3)(x + 4).

Worked example: Factor x² − 2x − 15.

• Need product −15 and sum −2 → −5 and 3.
• Factored form: (x − 5)(x + 3).

Factoring ax² + bx + c (a ≠ 1)

When the leading coefficient is not 1, use the AC method (factor by grouping). Multiply a·c, find two numbers that multiply to that product and add to b, split the middle term, then group.

Worked example: Factor 2x² + 7x + 3.

• a·c = 2·3 = 6. Find two numbers with product 6 and sum 7 → 6 and 1.
• Split the middle term: 2x² + 6x + x + 3.
• Group: 2x(x + 3) + 1(x + 3).
• Factor out (x + 3): (2x + 1)(x + 3).

Difference of Squares

The pattern a² − b² = (a + b)(a − b) factors any difference of two perfect squares. It requires a MINUS sign — a sum of squares does not factor over the real numbers.

Worked example: Factor x² − 49 = (x + 7)(x − 7).

Worked example: Factor 9x² − 25 = (3x + 5)(3x − 5), since 9x² = (3x)² and 25 = 5².

Rational Expressions and Restrictions

A rational expression is a fraction whose numerator and denominator are polynomials, such as (x² − 9)/(x² + 5x + 6). A restriction is any value of the variable that makes a denominator equal to zero; division by zero is undefined, so those values must be excluded from the domain.

Worked example (simplify): Simplify (x² − 9)/(x² + 5x + 6).

• Factor numerator: x² − 9 = (x + 3)(x − 3).
• Factor denominator: x² + 5x + 6 = (x + 2)(x + 3).
• Restrictions: denominator zero when x = −2 or x = −3.
• Cancel the common factor (x + 3): result is (x − 3)/(x + 2), with x ≠ −2 and x ≠ −3.

Multiplying and Dividing Rational Expressions

Multiply: factor everything, cancel common factors, then multiply across. Divide: multiply by the reciprocal of the second fraction (flip and multiply), then proceed as multiplication.

Worked example (multiply): (x/(x+2)) · ((x+2)/(x−1)). Cancel the common (x + 2) to get x/(x − 1), x ≠ −2, x ≠ 1.

Worked example (divide): ((x+3)/x) ÷ ((x+3)/(x−4)) = ((x+3)/x) · ((x−4)/(x+3)). Cancel (x + 3): result (x − 4)/x, x ≠ 0, x ≠ −4, x ≠ 4.

Common Mistakes

• Skipping the GCF step, which makes later factoring harder or incomplete.
• Trying to factor a SUM of squares (a² + b²) — it does not factor over the reals.
• Cancelling individual TERMS instead of whole FACTORS: you cannot cancel the x in (x + 3)/(x + 2).
• Forgetting to state restrictions, or stating them from the SIMPLIFIED form only (use the ORIGINAL denominators).
• Forgetting to flip the second fraction when dividing.

Factoring Out a Negative or Combining Methods

Sometimes the best first move is to factor out a negative GCF so the leading term becomes positive, making the trinomial easier to factor.

Worked example: Factor −x² + 5x − 6.

• Factor out −1: −(x² − 5x + 6).
• Factor the trinomial: x² − 5x + 6 = (x − 2)(x − 3).
• Final answer: −(x − 2)(x − 3).

Full factoring may also require a GCF step followed by a special pattern.

Worked example: Factor 2x² − 18.

• GCF is 2: 2(x² − 9).
• The inside is a difference of squares: x² − 9 = (x + 3)(x − 3).
• Final answer: 2(x + 3)(x − 3).

Adding and Subtracting Rational Expressions

Like numerical fractions, rational expressions need a common denominator before you add or subtract. Factor each denominator, build the least common denominator (LCD), rewrite each fraction, then combine the numerators.

Worked example: Simplify (3/x) + (2/x²).

• The LCD is x². Rewrite the first fraction: 3/x = 3x/x².
• Add: 3x/x² + 2/x² = (3x + 2)/x², with x ≠ 0.

Worked example (subtraction): Simplify (5/(x+1)) − (2/(x+1)). The denominators already match, so subtract numerators: (5 − 2)/(x + 1) = 3/(x + 1), with x ≠ −1.