IBEW Polynomials and Factoring 2 — Questions and Answers

Question 1: Factor completely: x² - 9

(x - 3)²
(x + 3)(x - 3) (Correct answer)
(x - 9)(x + 1)
(x + 9)(x - 1)

Correct answer: (x + 3)(x - 3)

This is a difference of squares: a² - b² = (a+b)(a-b). x² - 9 = x² - 3² = (x+3)(x-3).

x² - 9 = x² - 3². Using the difference of squares formula a² - b² = (a + b)(a - b): x² - 3² = (x + 3)(x - 3). Verify by FOILing: (x+3)(x-3) = x² - 3x + 3x - 9 = x² - 9 ✓.

Question 2: Multiply: (2x + 3)(x - 4)

2x² - 5x - 12 (Correct answer)
2x² - 5x + 12
2x² + 5x - 12
2x² - 8x - 12

Correct answer: 2x² - 5x - 12

FOIL: 2x·x + 2x·(-4) + 3·x + 3·(-4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12.

Using FOIL: First: 2x·x = 2x². Outer: 2x·(-4) = -8x. Inner: 3·x = 3x. Last: 3·(-4) = -12. Combine: 2x² + (-8x + 3x) - 12 = 2x² - 5x - 12.

Question 3: Factor: x² + 7x + 12

(x + 3)(x + 4) (Correct answer)
(x + 2)(x + 6)
(x + 1)(x + 12)
(x + 6)(x + 2)

Correct answer: (x + 3)(x + 4)

Find two numbers that multiply to 12 and add to 7: 3 and 4. So (x+3)(x+4).

For x² + bx + c, find two numbers that multiply to c=12 and add to b=7. Pairs multiplying to 12: (1,12), (2,6), (3,4). The pair (3,4) sums to 7 ✓. So x² + 7x + 12 = (x + 3)(x + 4).

Question 4: What is the GCF of 12x³y², 18x²y, and 24xy³?

6xy (Correct answer)
12xy
6x²y
18xy

Correct answer: 6xy

GCF of 12, 18, 24 is 6. Lowest power of x is x¹. Lowest power of y is y¹. GCF = 6xy.

Coefficients: GCF(12, 18, 24) = 6. Variable x: powers are 3, 2, 1 — lowest is x¹. Variable y: powers are 2, 1, 3 — lowest is y¹. GCF = 6xy.

Question 5: Factor completely: 3x² - 12

3(x² - 4)
3(x + 2)(x - 2) (Correct answer)
(3x + 6)(x - 2)
3(x - 2)²

Correct answer: 3(x + 2)(x - 2)

Factor out 3: 3(x² - 4). Then factor difference of squares: 3(x+2)(x-2).

Step 1: Factor out GCF of 3: 3x² - 12 = 3(x² - 4). Step 2: x² - 4 = x² - 2² is a difference of squares = (x+2)(x-2). Complete factorization: 3(x+2)(x-2).

Question 6: Simplify: (3x²y)(4xy³)

12x²y³
12x³y⁴ (Correct answer)
7x³y⁴
12x³y³

Correct answer: 12x³y⁴

Multiply coefficients: 3×4=12. Add exponents: x²·x¹=x³, y¹·y³=y⁴. Result: 12x³y⁴.

Multiply term by term: Coefficients: 3 × 4 = 12. x terms: x² × x¹ = x^(2+1) = x³. y terms: y¹ × y³ = y^(1+3) = y⁴. Final answer: 12x³y⁴.

IBEW Polynomials and Factoring 2

Factor completely: x² - 9

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