Rise Placement Test Exponents and Polynomials Questions and Answers — Questions and Answers
Question 1: Which of the following expressions is equivalent to (3x⁻⁴y⁵) / (x²y⁻²)?
- 3x⁻⁶y³
- 3x²y⁷
- 3y⁷ / x⁶ (Correct answer)
- 3y³ / x⁶
Correct answer: 3y⁷ / x⁶
To simplify the expression, use the quotient rule for exponents, which states that xᵃ/xᵇ = xᵃ⁻ᵇ. For the x terms: x⁻⁴ / x² = x⁻⁴⁻² = x⁻⁶. For the y terms: y⁵ / y⁻² = y⁵⁻⁽⁻²⁾ = y⁷. The expression becomes 3x⁻⁶y⁷. A negative exponent indicates a reciprocal, so x⁻⁶ moves to the denominator to become x⁶. The final simplified expression is 3y⁷ / x⁶.
Question 2: A rectangular park has a length of (4x + 5) meters and a width of (x - 2) meters. Which polynomial represents the total area of the park in square meters?
- 4x² - 3x - 10 (Correct answer)
- 4x² + 13x - 10
- 5x + 3
- 4x² - 10
Correct answer: 4x² - 3x - 10
The area of a rectangle is found by multiplying its length by its width. In this case, Area = (4x + 5)(x - 2). Use the FOIL method to multiply the binomials: First (4x * x = 4x²), Outer (4x * -2 = -8x), Inner (5 * x = 5x), Last (5 * -2 = -10). Combine the like terms: 4x² - 8x + 5x - 10 = 4x² - 3x - 10.
Question 3: Perform the subtraction: (8x² - 5x + 3) - (3x² + 2x - 7)
- 5x² - 3x - 4
- 5x² - 7x - 4
- 11x² - 3x - 4
- 5x² - 7x + 10 (Correct answer)
Correct answer: 5x² - 7x + 10
To subtract the polynomials, distribute the negative sign to each term in the second polynomial: (8x² - 5x + 3) - 3x² - 2x + 7. Then, combine the like terms: (8x² - 3x²) + (-5x - 2x) + (3 + 7). This simplifies to 5x² - 7x + 10.
Question 4: Simplify the expression (2a³b⁻⁴)² completely.
- 4a⁶ / b⁸ (Correct answer)
- 2a⁵ / b²
- 4a⁵b⁻²
- 2a⁶ / b⁸
Correct answer: 4a⁶ / b⁸
To simplify, apply the outer exponent to each factor inside the parentheses. For the coefficient: 2² = 4. For the variables, use the power rule (xᵃ)ᵇ = xᵃᵇ. So, (a³)² = a³*² = a⁶, and (b⁻⁴)² = b⁻⁴*² = b⁻⁸. The expression becomes 4a⁶b⁻⁸. To express with only positive exponents, move the base with the negative exponent to the denominator, making the exponent positive. This results in 4a⁶ / b⁸.
Question 5: Which of the following is the correct factorization of the polynomial 9x² - 49?
- (9x - 7)(x + 7)
- (3x - 7)²
- (3x - 7)(3x + 7) (Correct answer)
- This polynomial cannot be factored.
Correct answer: (3x - 7)(3x + 7)
This polynomial is a difference of two squares, which follows the pattern a² - b² = (a - b)(a + b). In this case, a² is 9x², so a = 3x. And b² is 49, so b = 7. Applying the formula, the factorization is (3x - 7)(3x + 7).
Question 6: Find the product of (2x - 1) and (x² + 3x - 5).
- 2x³ - x² + 3x + 5
- 2x³ + 5x² - 13x + 5 (Correct answer)
- 2x³ + 7x² - 7x + 5
- x² + 5x - 6
Correct answer: 2x³ + 5x² - 13x + 5
To multiply a binomial by a trinomial, distribute each term of the binomial to every term of the trinomial. First, multiply 2x by the trinomial: 2x(x² + 3x - 5) = 2x³ + 6x² - 10x. Next, multiply -1 by the trinomial: -1(x² + 3x - 5) = -x² - 3x + 5. Finally, combine the like terms from both results: 2x³ + (6x² - x²) + (-10x - 3x) + 5 = 2x³ + 5x² - 13x + 5.
Rise Placement Test Exponents and Polynomials Questions and Answers
Which of the following expressions is equivalent to (3x⁻⁴y⁵) / (x²y⁻²)?