Question

11) -4, -1, -3

A) f(x) = x + 8x + 19x + 15
B) f(x) x + 8x? + 19x + 4
C) f(x)=r: 8x? 1 27x + 12
D) f(x) – xº +872 + 19x + 12

Write a polynomial function of least degree with integral coefficients that has the given zeros.

Answer 1

The required polynomial is:

`f\left(x\right)=x^3+8x^2+19x+12`

Explanation:

Given the zeros

-4, -1, -3

Zeros can be written as:

x = -4, x = -1, x = -3

Finding the factors

x = -4

x +4 = 0

x = -1

x + 1 = 0

x = -3

x + 3 = 0

Thus, the factor are:

(x+4) (x+1) (x+3)

The original function can be computed by multiplying the factors

`\left(x+4\right)\left(x+1\right)\left(x+3\right)=0`

as

`\left(x+4\right)\left(x+1\right)=x^2+5x+4`

so the expression becomes

`\left(x^2+5x+4\right)\left(x+3\right)=0`

Distribute parentheses

`x^2x+x^2\cdot \:3+5xx+5x\cdot \:3+4x+4\cdot \:3=0`

`x^2x+3x^2+5xx+5\cdot \:\:3x+4x+4\cdot \:\:3=0`

`x^3+3x^2+5x^2+15x+4x+12=0`

Add similar elements:
`3x^2+5x^2=8x^2`

`x^3+8x^2+15x+4x+12 = 0`

Add similar elements:
`15x+4x=19x`

`x^3+8x^2+19x+12=0`

Thus, the required polynomial is:

`f\left(x\right)=x^3+8x^2+19x+12`

Note: your answer choices are a little ambiguous, but I have explained the entire procedure. Hopefully, it will clear your concept.