Free Rise Placement Test Exponents and Polynomials Questions and Answers — Questions and Answers
Question 1: Which of the following expressions is equivalent to (3x²y⁻³)⁻²?
- 9x⁴y⁶
- -9x⁴y⁶
- y⁶ / (9x⁴) (Correct answer)
- 1 / (6x⁴y⁶)
Correct answer: y⁶ / (9x⁴)
To simplify (3x²y⁻³)⁻², first distribute the outer exponent -2 to each factor inside the parentheses. This gives 3⁻²(x²)⁻²(y⁻³)⁻². Using the power of a power rule (multiply exponents), this becomes 3⁻²x⁻⁴y⁶. A negative exponent means to take the reciprocal of the base. So, 3⁻² becomes 1/3² = 1/9, and x⁻⁴ becomes 1/x⁴. The y⁶ term remains in the numerator. Combining these gives y⁶ / (9x⁴).
Question 2: What is the product of the binomials (2a - 5) and (3a + 4)?
- 6a² - 20
- 6a² - 7a - 20 (Correct answer)
- 6a² + 7a - 20
- 5a - 1
Correct answer: 6a² - 7a - 20
To multiply the two binomials, use the FOIL method (First, Outer, Inner, Last). First: (2a)(3a) = 6a². Outer: (2a)(4) = 8a. Inner: (-5)(3a) = -15a. Last: (-5)(4) = -20. Then, combine the like terms (the Outer and Inner products): 8a - 15a = -7a. The final product is 6a² - 7a - 20.
Question 3: A rectangular garden has a length represented by the expression (4x + 3) feet and a width represented by (x - 2) feet. Which polynomial represents the total area of the garden in square feet?
- 4x² + 5x - 6
- 5x + 1
- 4x² - 5x - 6 (Correct answer)
- 4x² - 6
Correct answer: 4x² - 5x - 6
The area of a rectangle is calculated by multiplying its length by its width. In this case, Area = (4x + 3)(x - 2). Using the FOIL method: First: (4x)(x) = 4x². Outer: (4x)(-2) = -8x. Inner: (3)(x) = 3x. Last: (3)(-2) = -6. Combine the like terms: -8x + 3x = -5x. The polynomial representing the area is 4x² - 5x - 6.
Question 4: Which of the following correctly classifies the polynomial 8x³ - 12x + 5 by its degree and number of terms?
- Quadratic Trinomial
- Cubic Binomial
- Quadratic Binomial
- Cubic Trinomial (Correct answer)
Correct answer: Cubic Trinomial
The degree of a polynomial is determined by the highest exponent of the variable, which is 3 in this case, making it 'cubic'. The number of terms is found by counting the parts separated by plus or minus signs. There are three terms (8x³, -12x, and 5), which makes it a 'trinomial'. Therefore, the polynomial is a cubic trinomial.
Question 5: Perform the subtraction: (7x² - 3x + 9) - (4x² + 2x - 5)
- 3x² - x + 4
- 3x² - 5x + 4
- 3x² - 5x + 14 (Correct answer)
- 11x² - x + 4
Correct answer: 3x² - 5x + 14
To subtract the polynomials, distribute the negative sign to each term in the second parenthesis: (7x² - 3x + 9) - 4x² - 2x + 5. Then, combine like terms: (7x² - 4x²) + (-3x - 2x) + (9 + 5). This simplifies to 3x² - 5x + 14.
Question 6: Simplify the rational expression: (x² - 9) / (x² + 7x + 12)
- (x - 3) / (x + 4) (Correct answer)
- (x + 3) / (x + 4)
- -9 / (7x + 12)
- Already simplified
Correct answer: (x - 3) / (x + 4)
To simplify the rational expression, factor both the numerator and the denominator. The numerator, x² - 9, is a difference of squares and factors to (x - 3)(x + 3). The denominator, x² + 7x + 12, factors to (x + 3)(x + 4). The expression becomes [(x - 3)(x + 3)] / [(x + 3)(x + 4)]. The common factor (x + 3) can be canceled from the numerator and denominator, leaving (x - 3) / (x + 4).
Free Rise Placement Test Exponents and Polynomials Questions and Answers
Which of the following expressions is equivalent to (3x²y⁻³)⁻²?