PRE CALCULUS Polynomials and Rational Functions — Questions and Answers
Question 1: What is the degree of the polynomial 3x⁴ - 2x² + x - 7?
- 3
- 4 (Correct answer)
- 7
- 2
Correct answer: 4
The degree of a polynomial is the highest power of the variable. Here, the highest exponent is 4 (from the term 3x⁴).
Question 2: What is the Remainder Theorem?
- Every polynomial has a remainder
- When a polynomial f(x) is divided by (x - c), the remainder equals f(c) (Correct answer)
- Remainders are always zero
- The theorem only applies to quadratics
Correct answer: When a polynomial f(x) is divided by (x - c), the remainder equals f(c)
The Remainder Theorem states that dividing f(x) by (x - c) gives a remainder equal to f(c). If f(c) = 0, then (x - c) is a factor.
Question 3: What is a vertical asymptote of a rational function?
- A horizontal line the graph approaches
- A vertical line x = a where the function approaches infinity, occurring where the denominator equals zero (Correct answer)
- The y-intercept
- A diagonal line on the graph
Correct answer: A vertical line x = a where the function approaches infinity, occurring where the denominator equals zero
Vertical asymptotes occur at x-values where the denominator of a rational function equals zero (and the numerator does not), causing the function to approach infinity.
Question 4: How many roots can a polynomial of degree n have?
- Exactly n
- At most n real roots (Correct answer)
- Exactly 2n
- At most 1
Correct answer: At most n real roots
The Fundamental Theorem of Algebra states a polynomial of degree n has exactly n roots (counting complex roots and multiplicity), but it has at most n real roots.
Question 5: What does the end behavior of a polynomial depend on?
- The constant term
- The leading coefficient and the degree of the polynomial (Correct answer)
- The number of terms
- The middle terms
Correct answer: The leading coefficient and the degree of the polynomial
End behavior (what happens as x → ±∞) is determined by the leading term. The degree determines the general shape, and the leading coefficient determines the direction.
Question 6: What is the Factor Theorem?
- All polynomials can be factored
- If f(c) = 0, then (x - c) is a factor of the polynomial f(x) (Correct answer)
- Factors are always integers
- The theorem applies only to quadratics
Correct answer: If f(c) = 0, then (x - c) is a factor of the polynomial f(x)
The Factor Theorem is a special case of the Remainder Theorem: if substituting c into f(x) gives 0, then (x - c) divides evenly into f(x).
PRE CALCULUS Polynomials and Rational Functions
What is the degree of the polynomial 3x⁴ - 2x² + x - 7?