Questions: Polynomial Functions: Degree and Leading Coefficient

Question 1

What is the degree of the polynomial p(x) = 5x³ - 2x⁴ + x - 7?

• A. 3
• B. 4
• C. 5
• D. 1

Answer: (B)

The degree is the highest power of x with a nonzero coefficient. Even though x⁴ is not the first term written, it is the highest-power term, so the degree is 4. This polynomial is not in standard form — the degree is determined by the highest power present, not the first term written.

Question 2

A degree-4 polynomial generally has exactly 4 x-intercepts.

• T. True
• F. False

Answer: False

The degree gives the maximum number of real x-intercepts, not the exact count. A degree-4 polynomial can have 0, 1, 2, 3, or 4 real x-intercepts. For example, f(x) = x⁴ + 1 has no real x-intercepts because x⁴ + 1 > 0 for all real x. Some intercepts may be complex (non-real) or repeated.

Question 3

What is the degree and leading coefficient of p(x) = (x + 2)(x - 1)(x + 3)? You do not need to fully expand.

Model answer: Degree: 3. Leading coefficient: 1.

The highest-degree term comes from multiplying the leading x from each factor: x · x · x = x³. The coefficient of x³ is 1 (since each factor contributes a coefficient of 1). With three linear factors multiplied together, the degree must be 3. You can read off both values without expanding the entire product.