Question

Find the polynomial function in standard form that has the zeros listed.

1(multiplicity 2), -2(multiplicity 3)

A. f(x) = x^5 + 4x^4 + x^3 - 10x^2 + 4x
B. f(x) = x^5 + 6x^4 + x^3 - 10x^2 - 4x + 8
C. f(x) = x^4 + 4x^2 + x^3 - 10x^2 - 4x + 8
D. f(x) = x^5 + 8x^4 + 25x^3 + 38x^2 + 28x + 8

Answer 1

To find the polynomial function in standard form with the given zeros, we need to consider the multiplicity of each zero.

The zero 1 has a multiplicity of 2, which means it appears twice as a root of the polynomial. Similarly, the zero -2 has a multiplicity of 3, meaning it appears three times as a root.

To construct the polynomial function, we can use the following steps:

1. For the zero 1 with a multiplicity of 2, we know that (x - 1) should appear twice as a factor in the polynomial. Therefore, we have (x - 1)^2 = x^2 - 2x + 1.

2. For the zero -2 with a multiplicity of 3, we know that (x + 2) should appear three times as a factor in the polynomial. Therefore, we have (x + 2)^3 = x^3 + 6x^2 + 12x + 8.

Now, we multiply these factors together to obtain the polynomial function:

f(x) = (x^2 - 2x + 1)(x^3 + 6x^2 + 12x + 8)

Simplifying this expression by expanding and combining like terms, we get:

f(x) = x^5 + 4x^4 + x^3 - 10x^2 - 4x + 8

Comparing this with the answer choices provided, we can see that the correct polynomial function in standard form is option C:

f(x) = x^4 + 4x^2 + x^3 - 10x^2 - 4x + 8

Explanation:

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Answer 2

C. f(x) = x^4 + 4x^4 + x^3 - 10x^2 - 4x + 8 ✓

Given :

• zeros : 1(multiplicity 2) , -2( multiplicity 3)

To find :

• Polynomial function in it's standard form

Solution :

Let the polynomial be P(x) ,

ATQ,

P(x) = [{(x-1)(x-1)}{(x+2)(x+2)(x+2)}]

P(x) = [{x²-x -x + 1}{x³+6x²+12x+8}]

P(x) = (x²-2x +1)(x³+6x²+12x+8)

P(x) = x⁵ + 6x⁴ +12x³ +8x² -2x⁴ -12x³ -24x² -16x + x³ + 6x² +12x + 8

P(x) = x⁵ + 6x⁴ -2x⁴ +12x³ -12x³ + x³ +8x² -24x² +6x² -16x +12x + 8

P(x) = x⁵ + 4x⁴ + x³ - 10x² - 4x + 8

Therefore, option (C). f(x) = x^4 + 4x^4 + x^3 - 10x^2 - 4x + 8 is correct ✓ .