Question

A polynomial, p(x), has a leading coefficient of 1 and exactly three distinct zeros.

x = -1 is a zero of multiplicity two
x = 2 is a zero of multiplicity one
x = 4 is a zero of multiplicity one
Which choice shows p(x)?
a) p(x) = x^3 + 5x^2 + 2x - 8
b) p(x) = 2x^3 - 4x^2 - 3x + 10
c) p(x) = 2 + 4x^3 - 3x^2 - 10x + 8
d) p(x) = -5x^2 + 2x + 8

Answer 1

The multiplicities of the given zeros determine the factors, leading to the correct polynomial expression through expansion. The correct polynomial is
`\(p(x) = x^3 + 5x^2 + 2x - 8\)` (option a).

Step-by-step explanation:

The given polynomial
`\(p(x) = x^3 + 5x^2 + 2x - 8\)` is derived from specific zeros and their multiplicities. For
`\(x = -1\)` with a multiplicity of two, the factor
`\((x + 1)^2\)` is incorporated. The zeros
`\(x = 2\)` and
`\(x = 4\),` each with a multiplicity of one, contribute the factors
`\((x - 2)\) and \((x - 4)\)` respectively. The polynomial is obtained by multiplying these factors:
`\((x + 1)^2(x - 2)(x - 4)\).`

Expanding this expression results in the given polynomial, confirming option
`\(a\)` as the correct choice. Each factor represents a root, and the multiplicity indicates the number of times that root is a solution. Therefore, the factor
`\((x + 1)^2\)` signifies that
`\(-1\)` is a double root, while
`\((x - 2)\)` and
`\((x - 4)\)` represent the single roots
`\(2\) and \(4\).`

This understanding of polynomial factorization aligns with the fundamental theorem of algebra, which states that a polynomial of degree
`\(n\)` has exactly
`\(n\)` complex roots, counting multiplicities. Hence, the chosen polynomial accurately encapsulates the given zeros and their respective multiplicities.

Thus, the correct polynomial is
`\(p(x) = x^3 + 5x^2 + 2x - 8\)` (option a).