IBEW Aptitude Test Polynomial Factoring Methods Questions and Answers — Questions and Answers
Question 1: An electrical calculation results in the polynomial 4x² - 49. Which of the following represents the correct factored form of this expression?
- (4x - 7)(x + 7)
- (2x - 7)²
- (2x + 7)(2x - 7) (Correct answer)
- This polynomial cannot be factored.
Correct answer: (2x + 7)(2x - 7)
This polynomial is a 'difference of squares' of the form a² - b². In this case, a² is 4x² (so a=2x) and b² is 49 (so b=7). The formula for factoring a difference of squares is (a + b)(a - b), which results in (2x + 7)(2x - 7).
Question 2: Which of the following expressions is the result of fully factoring 3x³ + 6x² - 9x?
- 3x(x² + 2x - 3)
- 3(x³ + 2x² - 3x)
- 3x(x + 3)(x - 1) (Correct answer)
- x(3x + 9)(x - 1)
Correct answer: 3x(x + 3)(x - 1)
The first step in factoring any polynomial is to find the Greatest Common Factor (GCF). The GCF of all terms is 3x. Factoring this out leaves 3x(x² + 2x - 3). The remaining trinomial (x² + 2x - 3) can be factored further. We need two numbers that multiply to -3 and add to +2. These numbers are +3 and -1. So, the trinomial factors to (x + 3)(x - 1). The fully factored form is 3x(x + 3)(x - 1).
Question 3: A signal processing equation is given by the polynomial 2x² - 5x - 12. What is its factored form?
- (2x + 3)(x - 4) (Correct answer)
- (2x - 4)(x + 3)
- (2x - 6)(x + 2)
- (x + 4)(2x - 3)
Correct answer: (2x + 3)(x - 4)
This is a trinomial where the leading coefficient is not 1. Using the 'AC' method, we multiply a*c, which is 2 * -12 = -24. We need two numbers that multiply to -24 and add to -5. These numbers are -8 and +3. We rewrite the middle term: 2x² - 8x + 3x - 12. Now, we factor by grouping: (2x² - 8x) + (3x - 12). Factor out the GCF from each group: 2x(x - 4) + 3(x - 4). The final factored form is (2x + 3)(x - 4).
Question 4: Factor the polynomial 9x² - 24x + 16 completely.
- (9x - 4)(x - 4)
- (3x - 4)² (Correct answer)
- (3x + 4)(3x - 4)
- (3x - 8)(3x - 2)
Correct answer: (3x - 4)²
This expression is a perfect square trinomial. The first term, 9x², is a perfect square (3x)². The last term, 16, is a perfect square (4)². The middle term, -24x, is -2 times the product of the square roots of the first and last terms (-2 * 3x * 4 = -24x). This fits the pattern a² - 2ab + b², which factors to (a - b)². Therefore, the factored form is (3x - 4)².
Question 5: An apprentice is trying to simplify the expression x³ - 4x² + 5x - 20. Which factoring method is most appropriate to start with for this polynomial?
- Difference of Squares
- Factoring by Grouping (Correct answer)
- Greatest Common Factor
- Perfect Square Trinomial
Correct answer: Factoring by Grouping
For a polynomial with four terms like this one, the most direct method to try is factoring by grouping. You group the first two terms and the last two terms: (x³ - 4x²) + (5x - 20). Then, you factor the GCF from each group: x²(x - 4) + 5(x - 4). This results in the factored form (x² + 5)(x - 4).
Question 6: To solve for a variable in a circuit, you need to factor the expression n² + 8n - 48. What is the correct factored form?
- (n + 6)(n - 8)
- (n - 4)(n + 12) (Correct answer)
- (n - 6)(n + 8)
- (n + 4)(n - 12)
Correct answer: (n - 4)(n + 12)
To factor the trinomial n² + 8n - 48, you need to find two numbers that multiply to -48 and add to +8. By listing the factors of 48, we can find the pair +12 and -4. Their product is -48 and their sum is +8. Therefore, the factored form is (n - 4)(n + 12).
IBEW Aptitude Test Polynomial Factoring Methods Questions and Answers
An electrical calculation results in the polynomial 4x² - 49.
Which of the following represents the correct factored form of this expression?