Multiple Choice Questions
Algebra: Polynomial and Rational Functions (Review)
Topic: Polynomial and Rational Functions
Grade: 11
Question 1:
Which of the following polynomials has a degree of 4?
a) 3x^2 – 4x + 2
b) x^4 – 5x^3 + 2x^2 – 7x + 1
c) 4x^3 + 2x^2 – 3x + 1
d) 2x^5 – 3x^2 + 5
Answer: b) x^4 – 5x^3 + 2x^2 – 7x + 1
Explanation: The degree of a polynomial is determined by the highest exponent of the variable. In this case, the highest exponent is 4, so the polynomial has a degree of 4. For example, if we let x = 2, the polynomial becomes 16 – 40 + 8 – 14 + 1 = -29.
Question 2:
Which of the following polynomials is a binomial?
a) x^2 – 3x + 1
b) 2x^3 – 5x^2 + 4x – 1
c) 3x – 2
d) x^4 + 2x^2 – 3x + 1
Answer: c) 3x – 2
Explanation: A binomial is a polynomial with exactly two terms. In this case, the polynomial 3x – 2 has two terms, so it is a binomial. For example, if we let x = 4, the polynomial becomes 12 – 2 = 10.
Question 3:
Which of the following is a factor of the polynomial x^3 – 4x^2 + 3x + 6?
a) x – 2
b) x + 2
c) x – 3
d) x + 3
Answer: d) x + 3
Explanation: To determine if a polynomial is a factor of another polynomial, we can use synthetic division. If the remainder is 0, then the polynomial is a factor. In this case, when we divide x^3 – 4x^2 + 3x + 6 by x + 3, we get a remainder of 0. For example, if we let x = 1, the polynomial becomes 1 – 4 + 3 + 6 = 6.
Question 4:
Which of the following is a root of the equation x^2 + 5x – 6 = 0?
a) -3
b) 2
c) -2
d) 3
Answer: a) -3
Explanation: To find the roots of a quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 – 4ac))/(2a). In this case, when we substitute the values a = 1, b = 5, and c = -6 into the quadratic formula, we get x = (-5 ± √(25 + 24))/2 = (-5 ± √49)/2 = (-5 ± 7)/2. Therefore, the roots are x = -3 and x = 1. For example, if we let x = -3, the equation becomes 9 – 15 – 6 = 0.
Question 5:
Which of the following rational functions has a vertical asymptote at x = 2?
a) f(x) = (x + 3)/(x – 2)
b) f(x) = (x – 2)/(x + 3)
c) f(x) = (x – 2)/(x – 2)
d) f(x) = (x + 2)/(x – 2)
Answer: a) f(x) = (x + 3)/(x – 2)
Explanation: A rational function has a vertical asymptote at x = a if the denominator of the function equals zero at x = a. In this case, the denominator of f(x) = (x + 3)/(x – 2) equals zero when x = 2. For example, if we let x = 2, the function becomes undefined.
Question 6:
Which of the following rational functions has a horizontal asymptote at y = 1?
a) f(x) = (2x^2 + 3x – 1)/(x^2 + 1)
b) f(x) = (x^2 + 3x – 1)/(2x^2 + 1)
c) f(x) = (x^2 + 3x – 1)/(x^2 – 1)
d) f(x) = (2x^2 + 3x – 1)/(2x^2 – 1)
Answer: a) f(x) = (2x^2 + 3x – 1)/(x^2 + 1)
Explanation: A rational function has a horizontal asymptote at y = a if the degrees of the numerator and denominator are the same and the leading coefficients are equal. In this case, the degrees are both 2 and the leading coefficients are both 2, so the horizontal asymptote is y = 1. For example, if we let x approach infinity, the function approaches 2x^2/x^2 = 2, which is equal to the horizontal asymptote.
Question 7:
Which of the following rational functions has a slant asymptote at y = x + 2?
a) f(x) = (x^2 + 3x + 1)/(x – 2)
b) f(x) = (x^2 + 2x + 1)/(x + 1)
c) f(x) = (x^2 + 4x + 3)/(x + 1)
d) f(x) = (x^2 + 3x + 1)/(x + 2)
Answer: b) f(x) = (x^2 + 2x + 1)/(x + 1)
Explanation: A rational function has a slant asymptote if the degree of the numerator is one more than the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 1, so the rational function has a slant asymptote. The equation of the slant asymptote is given by the division of the numerator by the denominator using long division or synthetic division. In this case, when we divide (x^2 + 2x + 1)/(x + 1), we get x + 1 as the quotient and the remainder is 0. Therefore, the slant asymptote is y = x + 1. For example, if we let x = 2, the function approaches 2 + 1 = 3, which is equal to the slant asymptote.
Question 8:
Which of the following rational functions has a hole at x = 2?
a) f(x) = (x^2 – 4)/(x – 2)
b) f(x) = (x^2 – 4)/(x + 2)
c) f(x) = (x^2 – 4)/(x – 4)
d) f(x) = (x^2 – 4)/(x + 4)
Answer: a) f(x) = (x^2 – 4)/(x – 2)
Explanation: A rational function has a hole at x = a if both the numerator and denominator have a common factor of (x – a). In this case, the numerator and denominator of f(x) = (x^2 – 4)/(x – 2) have a common factor of (x – 2). When we cancel out this common factor, we are left with f(x) = x + 2. Therefore, the rational function has a hole at x = 2. For example, if we let x = 2, the function becomes undefined.
Question 9:
Which of the following is the correct graph of the function f(x) = x^3 – 2x^2 + x – 1?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: c) Graph C
Explanation: The graph of a cubic function has one hump or two humps depending on the number of real roots. In this case, the function f(x) = x^3 – 2x^2 + x – 1 has one real root, so the graph should have one hump. Graph C is the only graph that matches this description. For example, if we let x = 0, the function becomes -1, which matches the y-intercept of Graph C.
Question 10:
Which of the following is the correct graph of the function f(x) = (x – 1)/(x + 2)?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: b) Graph B
Explanation: The graph of a rational function has a vertical asymptote at x = a if the denominator equals zero at x = a. In this case, the denominator of f(x) = (x – 1)/(x + 2) equals zero when x = -2, so the graph should have a vertical asymptote at x = -2. Graph B is the only graph that matches this description. For example, if we let x = -2, the function becomes undefined.
Question 11:
Which of the following is the correct graph of the function f(x) = (x^2 – 4)/(x + 1)?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: c) Graph C
Explanation: The graph of a rational function has a horizontal asymptote at y = a if the degrees of the numerator and denominator are the same and the leading coefficients are equal. In this case, the degrees are both 2 and the leading coefficients are both 1, so the horizontal asymptote is y = 1. Graph C is the only graph that matches this description. For example, if we let x approach infinity, the function approaches (x^2)/x = x, which is equal to the horizontal asymptote.
Question 12:
Which of the following is the correct graph of the function f(x) = (x – 2)/(x^2 – 4)?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: a) Graph A
Explanation: The graph of a rational function has a hole at x = a if both the numerator and denominator have a common factor of (x – a). In this case, the numerator and denominator of f(x) = (x – 2)/(x^2 – 4) have a common factor of (x – 2). When we cancel out this common factor, we are left with f(x) = 1/(x + 2). Therefore, the rational function has a hole at x = 2. Graph A is the only graph that matches this description. For example, if we let x = 2, the function becomes undefined.
Question 13:
Which of the following is the correct graph of the function f(x) = x^2 – 2x + 1?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: a) Graph A
Explanation: The graph of a quadratic function is a parabola. In this case, the function f(x) = x^2 – 2x + 1 is a perfect square trinomial, which means the graph is a parabola that opens upwards. Graph A is the only graph that matches this description. For example, if we let x = 0, the function becomes 1, which matches the y-intercept of Graph A.
Question 14:
Which of the following is the correct graph of the function f(x) = x^2 + 2x + 1?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: a) Graph A
Explanation: The graph of a quadratic function is a parabola. In this case, the function f(x) = x^2 + 2x + 1 is a perfect square trinomial, which means the graph is a parabola that opens upwards. Graph A is the only graph that matches this description. For example, if we let x = 0, the function becomes 1, which matches the y-intercept of Graph A.
Question 15:
Which of the following is the correct graph of the function f(x) = x^2 – 4x + 4?
a) Graph A
b) Graph B
c) Graph C
d) Graph D
Answer: d) Graph D
Explanation: The graph of a quadratic function is a parabola. In this case, the function f(x) = x^2 – 4x + 4 is a perfect square trinomial, which means the graph is a parabola that opens upwards. Graph D is the only graph that matches this description. For example, if we let x = 2, the function becomes 4 – 8 + 4 = 0.