Answer :
To find a quartic polynomial function in standard form with given zeros [tex]\(2, -4, -2, \)[/tex] and [tex]\(-3\)[/tex], follow these steps:
1. Write Factors from Zeros:
Each zero corresponds to a factor. If the polynomial has zeros at 2, -4, -2, and -3, the corresponding factors will be:
- [tex]\( (x - 2) \)[/tex]
- [tex]\( (x + 4) \)[/tex]
- [tex]\( (x + 2) \)[/tex]
- [tex]\( (x + 3) \)[/tex]
2. Multiply the Factors:
To get the polynomial, multiply these factors together:
- First, multiply the first two factors:
[tex]\[
(x - 2)(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8
\][/tex]
- Then, multiply the next two factors:
[tex]\[
(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
\][/tex]
3. Combine the Results:
Now multiply the two quadratic expressions you found:
[tex]\[
(x^2 + 2x - 8)(x^2 + 5x + 6)
\][/tex]
- First, multiply [tex]\(x^2\)[/tex] by every term in the second parentheses:
[tex]\[
x^2 \cdot (x^2 + 5x + 6) = x^4 + 5x^3 + 6x^2
\][/tex]
- Next, multiply [tex]\(2x\)[/tex] by every term in the second parentheses:
[tex]\[
2x \cdot (x^2 + 5x + 6) = 2x^3 + 10x^2 + 12x
\][/tex]
- Then, multiply [tex]\(-8\)[/tex] by every term in the second parentheses:
[tex]\[
-8 \cdot (x^2 + 5x + 6) = -8x^2 - 40x - 48
\][/tex]
4. Add Up All the Terms:
Now combine all these products:
[tex]\[
x^4 + 5x^3 + 6x^2 + 2x^3 + 10x^2 + 12x - 8x^2 - 40x - 48
\][/tex]
Combine like terms:
- [tex]\(x^4\)[/tex]
- [tex]\(5x^3 + 2x^3 = 7x^3\)[/tex]
- [tex]\(6x^2 + 10x^2 - 8x^2 = 8x^2\)[/tex]
- [tex]\(12x - 40x = -28x\)[/tex]
- [tex]\(-48\)[/tex]
So, the polynomial in standard form is:
[tex]\[
g(x) = x^4 + 7x^3 + 8x^2 - 28x - 48
\][/tex]
This matches option c, which is the correct answer.
1. Write Factors from Zeros:
Each zero corresponds to a factor. If the polynomial has zeros at 2, -4, -2, and -3, the corresponding factors will be:
- [tex]\( (x - 2) \)[/tex]
- [tex]\( (x + 4) \)[/tex]
- [tex]\( (x + 2) \)[/tex]
- [tex]\( (x + 3) \)[/tex]
2. Multiply the Factors:
To get the polynomial, multiply these factors together:
- First, multiply the first two factors:
[tex]\[
(x - 2)(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8
\][/tex]
- Then, multiply the next two factors:
[tex]\[
(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
\][/tex]
3. Combine the Results:
Now multiply the two quadratic expressions you found:
[tex]\[
(x^2 + 2x - 8)(x^2 + 5x + 6)
\][/tex]
- First, multiply [tex]\(x^2\)[/tex] by every term in the second parentheses:
[tex]\[
x^2 \cdot (x^2 + 5x + 6) = x^4 + 5x^3 + 6x^2
\][/tex]
- Next, multiply [tex]\(2x\)[/tex] by every term in the second parentheses:
[tex]\[
2x \cdot (x^2 + 5x + 6) = 2x^3 + 10x^2 + 12x
\][/tex]
- Then, multiply [tex]\(-8\)[/tex] by every term in the second parentheses:
[tex]\[
-8 \cdot (x^2 + 5x + 6) = -8x^2 - 40x - 48
\][/tex]
4. Add Up All the Terms:
Now combine all these products:
[tex]\[
x^4 + 5x^3 + 6x^2 + 2x^3 + 10x^2 + 12x - 8x^2 - 40x - 48
\][/tex]
Combine like terms:
- [tex]\(x^4\)[/tex]
- [tex]\(5x^3 + 2x^3 = 7x^3\)[/tex]
- [tex]\(6x^2 + 10x^2 - 8x^2 = 8x^2\)[/tex]
- [tex]\(12x - 40x = -28x\)[/tex]
- [tex]\(-48\)[/tex]
So, the polynomial in standard form is:
[tex]\[
g(x) = x^4 + 7x^3 + 8x^2 - 28x - 48
\][/tex]
This matches option c, which is the correct answer.
