Fundamentals of Polynomials

Author: Justice

This quiz covers basic concepts of polynomials, including degree, leading coefficient, constant term, operations on polynomials, and key theorems such as Bézout’s theorem.

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1 What correctly defines a polynomial of degree 3?

A An expression whose highest power of the variable is 3 B An expression that has exactly three terms C An expression that equals 3 when x = 1 D An expression that contains three different variables

2 What is the leading coefficient of W(x) = -2x⁴ + 5x³ - x + 7?

A 7, because it is the constant term in the polynomial B -1, because it is the coefficient of the linear term C -2, because it is the coefficient of the highest power D 5, because it is the coefficient of the second highest power

3 For P(x) = 3x³ - 4x² + 2x - 9, which value is the constant term?

A 3, since it is the coefficient of the cubic term B -4, since it is the coefficient of the quadratic term C -9, since it is the term independent of x D 2, since it is the coefficient of the linear term

4 Which description best characterizes the zero polynomial?

A A polynomial that has value 0 only when x = 0 B A polynomial in which all coefficients are equal to 0 C A polynomial whose degree is exactly zero D A polynomial that has only the root x = 0

5 What is the degree of W(x) = (x - 1)(x + 2)(x - 3)?

A 1, because each factor individually has degree 1 B 2, because the polynomial has three linear factors C 3, because the sum of the degrees of the factors is 3 D 6, because the product of degrees of the factors is 6

6 What is P(x) + Q(x) for P(x) = 3x³ - 2x² + 5x - 1 and Q(x) = -x³ + 4x² - 3x + 7?

A x³ + 2x² + 5x + 6 B 4x³ + 2x² + 2x - 8 C 2x³ + 2x² + 2x + 6 D 2x³ + x² + 8x - 8

7 What is P(x) - Q(x) for P(x) = 5x⁴ - 3x² + 2 and Q(x) = 2x⁴ + x³ - 3x² - 4?

A 3x⁴ - x³ + 6 B 7x⁴ - x³ - 1 C 3x⁴ - x³ - 3x² - 2 D 3x⁴ + x³ - 6

8 What is the product P(x)·Q(x) for P(x) = 2x² - 3x + 1 and Q(x) = x + 4?

A 2x³ + 5x² - 7x - 4 B 2x³ + 8x² - 3x + 1 C 2x³ + 5x² - 11x + 4 D 2x³ + x² - 11x + 4

9 When dividing P(x) = 2x³ - 5x² + 3x - 7 by Q(x) = x - 2 using Horner’s method, what is the remainder?

A 5, because the signs from Horner’s method must be reversed B -5, because it is the final value in the bottom row C 0, because the divisor is linear in x D 2, because the last coefficient in the scheme is 2

10 What is W(3) if W(x) = x⁴ - 2x³ + 3x² - x + 5?

A 56, because 81 - 54 + 27 - 3 + 5 = 56 B 44, because 81 - 54 + 27 - 3 - 7 = 44 C 60, because 81 - 54 + 27 + 1 + 5 = 60 D 50, because 81 - 54 + 27 - 3 - 1 = 50

11 Why is the statement “The sum of two polynomials of degree 4 is always a polynomial of degree 4” false?

A Because the sum can have degree greater than 4 after combining terms B Because the leading terms can cancel, giving a polynomial of lower degree C Because adding polynomials always lowers the degree by at least one D Because the sum of two degree 4 polynomials is never defined

12 What is the degree of the product of a polynomial of degree 2 and a polynomial of degree 3?

A 6, because the degrees multiply when taking the product B 3, because the lower degree is dominant in multiplication C 2, because the product cannot exceed the higher degree D 5, because the degrees add when multiplying polynomials

13 Which statement correctly describes Bézout’s theorem for polynomials?

A If P(a) = 0, then P(x) is divisible by (x - a) B If P(a) = 0, then P(x) is divisible by (x + a) C If P(a) = 0, then P(x) has no linear factors in x D If P(a) ≠ 0, then P(x) is divisible by (x - a)

14 What is true about the number of real roots of a polynomial of degree n?

A It always has exactly n distinct real roots B It has at most n real roots, possibly fewer C It always has at least one non-real complex root D It must have exactly n real roots counting multiplicity

15 How is the constant term of a polynomial P(x) related to the value P(0)?

A The constant term equals P(0) only if the polynomial has degree 1 B The constant term is unrelated to P(0) for polynomials of degree ≥ 2 C The constant term equals P(0) only when there is no linear term D The constant term equals P(0) because all powers of x become 0

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