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Quiz: Polynomial Functions

Test your understanding of polynomial functions, end behavior, division techniques, and root finding with these review questions.

1. What is the degree of the polynomial \(f(x) = 4x^5 - 2x^3 + 7x^2 - 1\)?

  1. \(4\)
  2. \(5\)
  3. \(3\)
  4. \(10\)
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The correct answer is B. The degree of a polynomial is the highest power of the variable that appears. Here, \(x^5\) is the highest power, so the degree is \(5\). Option A mistakes the leading coefficient for the degree. The degree tells us important structural information: a degree-\(5\) polynomial can have up to \(5\) real roots and up to \(4\) turning points.

Concept Tested: Degree of a Polynomial

2. Describe the end behavior of \(f(x) = -3x^4 + 2x^2 - 1\) as \(x \to \infty\).

  1. \(f(x) \to \infty\)
  2. \(f(x) \to 0\)
  3. \(f(x) \to -\infty\)
  4. \(f(x)\) oscillates
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The correct answer is C. The degree is \(4\) (even) and the leading coefficient is \(-3\) (negative). For even degree with negative leading coefficient, both ends of the graph go down, so as \(x \to \infty\), \(f(x) \to -\infty\) (and same as \(x \to -\infty\)). Only the leading term matters for end behavior because it dominates all lower-order terms for large \(|x|\).

Concept Tested: End Behavior

3. The factor theorem states that \((x - c)\) is a factor of \(f(x)\) if and only if:

  1. \(f(0) = c\)
  2. \(f(c) = 0\)
  3. \(f(c) = c\)
  4. \(f(1) = c\)
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The correct answer is B. The factor theorem says that \((x - c)\) divides \(f(x)\) evenly precisely when \(c\) is a root of \(f\), i.e., \(f(c) = 0\). This is enormously useful: once you find a root by inspection or trial, you can factor out \((x - c)\) and reduce the polynomial's degree, making further factoring easier. The factor theorem is a special case of the remainder theorem.

Concept Tested: Factor Theorem

4. What is the maximum number of turning points a polynomial of degree \(6\) can have?

  1. \(5\)
  2. \(6\)
  3. \(7\)
  4. \(3\)
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The correct answer is A. A polynomial of degree \(n\) can have at most \(n - 1\) turning points. For degree \(6\): \(6 - 1 = 5\). The actual number can be less, but never more. This "one less than the degree" rule is a direct consequence of calculus (the derivative has one less degree). Use this fact to sanity-check your sketches of polynomial graphs.

Concept Tested: Turning Points Count

5. What is the remainder when \(f(x) = x^3 - 2x^2 + 3x - 5\) is divided by \((x - 2)\)?

  1. \(-3\)
  2. \(5\)
  3. \(1\)
  4. \(0\)
Show Answer

The correct answer is C. By the remainder theorem, the remainder when dividing \(f(x)\) by \((x - c)\) is \(f(c)\). Here, \(f(2) = 8 - 8 + 6 - 5 = 1\). Since the remainder is not zero, \((x - 2)\) is not a factor. The remainder theorem is a huge time-saver — you avoid performing the full division just to find the remainder.

Concept Tested: Remainder Theorem

6. For the polynomial \(f(x) = (x - 2)^2(x + 3)\), what happens at \(x = 2\)?

  1. The graph crosses the \(x\)-axis
  2. The graph touches the \(x\)-axis and bounces back
  3. The graph has a vertical asymptote
  4. The graph has a hole
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The correct answer is B. The factor \((x - 2)^2\) indicates that \(x = 2\) is a root with multiplicity \(2\). Roots with even multiplicity cause the graph to touch the \(x\)-axis and bounce back rather than cross. Roots with odd multiplicity cause crossing. Holes and asymptotes are features of rational functions, not polynomials — polynomials are continuous and smooth everywhere.

Concept Tested: Multiplicity of Roots

7. The roots of \(x^3 + 2x^2 - 5x - 6 = 0\) sum to what value (using the sum of roots formula)?

  1. \(2\)
  2. \(6\)
  3. \(-6\)
  4. \(-2\)
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The correct answer is D. For a polynomial \(a_n x^n + a_{n-1}x^{n-1} + \cdots\), the sum of roots is \(-\frac{a_{n-1}}{a_n}\). Here \(a_n = 1\) (coefficient of \(x^3\)) and \(a_{n-1} = 2\) (coefficient of \(x^2\)). So the sum is \(-\frac{2}{1} = -2\). This formula lets you find the sum of roots without actually solving the polynomial.

Concept Tested: Sum of Roots Formula

8. Which of the following describes a cubic function?

  1. Degree \(2\), up to \(1\) turning point
  2. Degree \(4\), up to \(3\) turning points
  3. Degree \(3\), up to \(2\) turning points
  4. Degree \(5\), up to \(4\) turning points
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The correct answer is C. A cubic function has degree \(3\), and by the rule "at most \(n - 1\) turning points," it can have up to \(2\) turning points. Cubics always have at least one real root (the graph must cross the \(x\)-axis at least once because of the opposing end behavior of odd-degree polynomials). Option A describes a quadratic, B describes a quartic, and D describes a quintic.

Concept Tested: Cubic Function

9. Given that \(x = -1\) is a root of \(f(x) = x^3 - 4x^2 + x + 6\), what are the remaining two roots?

  1. \(x = 2\) and \(x = 3\)
  2. \(x = -2\) and \(x = 3\)
  3. \(x = 2\) and \(x = -3\)
  4. \(x = -2\) and \(x = -3\)
Show Answer

The correct answer is A. First verify: \(f(-1) = -1 - 4 - 1 + 6 = 0\). ✓ Divide \(f(x)\) by \((x + 1)\) using synthetic division with coefficients \([1, -4, 1, 6]\) and \(c = -1\): the quotient is \(x^2 - 5x + 6\). Factor: \((x - 2)(x - 3) = 0\), giving \(x = 2\) and \(x = 3\). So the three roots are \(-1, 2, 3\).

Concept Tested: Polynomial Division

10. What is the leading coefficient of \(f(x) = 5 - 3x + 2x^4 - x^2\)?

  1. \(5\)
  2. \(-1\)
  3. \(-3\)
  4. \(2\)
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The correct answer is D. The leading coefficient is the coefficient of the highest-power term. First rewrite in standard order: \(f(x) = 2x^4 - x^2 - 3x + 5\). The highest power is \(x^4\), with coefficient \(2\). A common mistake is to pick the first coefficient you see (option A picks the constant term) without rearranging. Always reorder by descending powers first.

Concept Tested: Leading Coefficient