Answer :
To form a polynomial using the given zeros, we must first realize that the zeros of the polynomial will be set as the roots of its factors. The zeros given are [tex]\(-5\)[/tex], [tex]\(-4\)[/tex], [tex]\(-3\)[/tex], and [tex]\(3\)[/tex], and the polynomial should have a degree of 4. To create the polynomial, follow these steps:
1. Write Factors for Each Zero: For each zero, there is a corresponding factor of the polynomial. If a number [tex]\(a\)[/tex] is a zero of the polynomial, then [tex]\((x - a)\)[/tex] is a factor. So, the factors are:
- [tex]\(x + 5\)[/tex] for the zero [tex]\(-5\)[/tex]
- [tex]\(x + 4\)[/tex] for the zero [tex]\(-4\)[/tex]
- [tex]\(x + 3\)[/tex] for the zero [tex]\(-3\)[/tex]
- [tex]\(x - 3\)[/tex] for the zero [tex]\(3\)[/tex]
2. Multiply the Factors Together: The polynomial is formed by multiplying these factors together:
[tex]\[
(x + 5)(x + 4)(x + 3)(x - 3)
\][/tex]
3. Expand the Polynomial: To express the polynomial in standard form, expand these factors:
- Start by multiplying two factors at a time:
[tex]\[
(x + 5)(x + 4) = x^2 + 9x + 20
\][/tex]
[tex]\[
(x + 3)(x - 3) = x^2 - 9
\][/tex]
- Now multiply these results:
[tex]\[
(x^2 + 9x + 20)(x^2 - 9)
\][/tex]
4. Final Step of Expansion: Expand the resulting expression:
[tex]\[
x^2(x^2 - 9) + 9x(x^2 - 9) + 20(x^2 - 9)
\][/tex]
- Calculate these terms:
[tex]\[
x^2 \cdot x^2 - 9x^2 = x^4 - 9x^2
\][/tex]
[tex]\[
9x \cdot x^2 - 9 \cdot 9x = 9x^3 - 81x
\][/tex]
[tex]\[
20 \cdot x^2 - 20 \cdot 9 = 20x^2 - 180
\][/tex]
5. Combine All Terms: Add all the results:
[tex]\[
x^4 + 9x^3 + 11x^2 - 81x - 180
\][/tex]
So, the polynomial that has zeros [tex]\(-5\)[/tex], [tex]\(-4\)[/tex], [tex]\(-3\)[/tex], and [tex]\(3\)[/tex], with a leading coefficient of 1 and is of degree 4, is:
[tex]\[
x^4 + 9x^3 + 11x^2 - 81x - 180
\][/tex]
Therefore, the correct polynomial from the given options is:
[tex]\[
x^4 + 9x^3 + 11x^2 - 81x - 180
\][/tex]
1. Write Factors for Each Zero: For each zero, there is a corresponding factor of the polynomial. If a number [tex]\(a\)[/tex] is a zero of the polynomial, then [tex]\((x - a)\)[/tex] is a factor. So, the factors are:
- [tex]\(x + 5\)[/tex] for the zero [tex]\(-5\)[/tex]
- [tex]\(x + 4\)[/tex] for the zero [tex]\(-4\)[/tex]
- [tex]\(x + 3\)[/tex] for the zero [tex]\(-3\)[/tex]
- [tex]\(x - 3\)[/tex] for the zero [tex]\(3\)[/tex]
2. Multiply the Factors Together: The polynomial is formed by multiplying these factors together:
[tex]\[
(x + 5)(x + 4)(x + 3)(x - 3)
\][/tex]
3. Expand the Polynomial: To express the polynomial in standard form, expand these factors:
- Start by multiplying two factors at a time:
[tex]\[
(x + 5)(x + 4) = x^2 + 9x + 20
\][/tex]
[tex]\[
(x + 3)(x - 3) = x^2 - 9
\][/tex]
- Now multiply these results:
[tex]\[
(x^2 + 9x + 20)(x^2 - 9)
\][/tex]
4. Final Step of Expansion: Expand the resulting expression:
[tex]\[
x^2(x^2 - 9) + 9x(x^2 - 9) + 20(x^2 - 9)
\][/tex]
- Calculate these terms:
[tex]\[
x^2 \cdot x^2 - 9x^2 = x^4 - 9x^2
\][/tex]
[tex]\[
9x \cdot x^2 - 9 \cdot 9x = 9x^3 - 81x
\][/tex]
[tex]\[
20 \cdot x^2 - 20 \cdot 9 = 20x^2 - 180
\][/tex]
5. Combine All Terms: Add all the results:
[tex]\[
x^4 + 9x^3 + 11x^2 - 81x - 180
\][/tex]
So, the polynomial that has zeros [tex]\(-5\)[/tex], [tex]\(-4\)[/tex], [tex]\(-3\)[/tex], and [tex]\(3\)[/tex], with a leading coefficient of 1 and is of degree 4, is:
[tex]\[
x^4 + 9x^3 + 11x^2 - 81x - 180
\][/tex]
Therefore, the correct polynomial from the given options is:
[tex]\[
x^4 + 9x^3 + 11x^2 - 81x - 180
\][/tex]
