Factoring and Polynomials: Practice Questions & Explanations

Factoring a trinomial x² + bx + c: find two numbers that multiply to c and add to b. Here: multiply to 12, add to 7. Try pairs: 1 × 12 = 12, but 1 + 12 = 13 ✗; 2 × 6 = 12, but 2 + 6 = 8 ✗; 3 × 4 = 12, and 3 + 4 = 7 ✓. So x² + 7x + 12 = (x + 3)(x + 4). Verify by FOIL (First, Outer, Inner, Last): (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓. Signs guide factoring: (1) c positive, b positive — both factors positive; (2) c positive, b negative — both factors negative; (3) c negative — factors have opposite signs (the larger absolute value matches the sign of b). Example with negative signs: x² − 5x + 6 = (x − 2)(x − 3); x² + x − 6 = (x + 3)(x − 2). Factoring is the reverse of FOIL. Used to solve quadratic equations: set the factored form = 0 and use zero-product property.

Difference of squares pattern: a² − b² = (a + b)(a − b). Here x² − 16 = x² − 4², so a = x and b = 4. Factored: (x + 4)(x − 4). Verify by FOIL: (x + 4)(x − 4) = x² − 4x + 4x − 16 = x² − 16 ✓. The middle terms cancel because of the +/− pattern. Key pattern recognition: a² − b² (difference) factors easily; a² + b² (sum) does NOT factor over real numbers (factors using complex numbers as (a + bi)(a − bi)). Common applications: (1) x² − 25 = (x + 5)(x − 5); (2) 4x² − 9 = (2x + 3)(2x − 3); (3) x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2) — factor repeatedly. Perfect square trinomial patterns to also recognize: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Example: x² + 6x + 9 = (x + 3)².

Solve quadratic equations by factoring when possible. Step 1: factor — find numbers that multiply to 6 and add to −5: that's −2 and −3 (since (−2)(−3) = 6 and −2 + −3 = −5). So x² − 5x + 6 = (x − 2)(x − 3). Step 2: set the factored form equal to zero: (x − 2)(x − 3) = 0. Step 3: zero-product property — if a product is zero, at least one factor is zero. So x − 2 = 0 (giving x = 2) or x − 3 = 0 (giving x = 3). Solutions: x = 2 or x = 3. Verify: 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓; 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓. A quadratic equation can have 0, 1, or 2 real solutions. If factoring doesn't work, use the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0. The discriminant b² − 4ac: positive → 2 solutions; zero → 1 solution; negative → no real solutions.

Factor and cancel. Numerator: x² − 9 = (x + 3)(x − 3) (difference of squares). Denominator: x − 3. Rewrite: [(x + 3)(x − 3)] / (x − 3). Cancel the (x − 3) factor: x + 3. The simplification is valid for x ≠ 3 (because at x = 3, the original expression has 0/0, undefined). For all other values of x, (x² − 9)/(x − 3) = x + 3. Strategy for simplifying rational expressions: (1) Factor numerator and denominator completely; (2) Cancel common factors; (3) State any restrictions on the variable (values that would make the original denominator zero). Common simplifications: (1) (x² − 4)/(x + 2) = (x + 2)(x − 2)/(x + 2) = x − 2; (2) (x² + 5x + 6)/(x + 2) = (x + 2)(x + 3)/(x + 2) = x + 3; (3) (x² − 25)/(x² + 10x + 25) = (x − 5)(x + 5)/(x + 5)² = (x − 5)/(x + 5). Don't cancel terms inside a sum — only factors of products. (x + 3)/(x + 5) does NOT simplify to 3/5 — those are terms, not factors.

To factor x² + 5x + 6, find two numbers that multiply to the constant (6) and add to the middle coefficient (5). Those numbers are 2 and 3, since 2 × 3 = 6 and 2 + 3 = 5. So the factors are (x + 2)(x + 3). You can verify by expanding: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. ✓ The other options either multiply to the wrong constant or add to the wrong middle term. The product-and-sum method is the standard approach for factoring simple quadratics.

First distribute the 3 across the parentheses: 3 × 2x = 6x and 3 × 4 = 12, giving 6x + 12. Then subtract 2x: 6x + 12 − 2x. Combine like terms (6x − 2x = 4x): the result is 4x + 12. The key steps are distributing first, then combining like terms (the x-terms together, the constants separately). A common error is forgetting to distribute the 3 to both terms inside the parentheses. The simplified expression is 4x + 12.