Answer :
To find the zeros of the polynomial [tex]\(9x^4 - 9x^2 - 4x^2 + 4\)[/tex] by grouping, we first simplify the expression:
1. Simplify: Combine like terms in the polynomial:
[tex]\[
9x^4 - (9x^2 + 4x^2) + 4 = 9x^4 - 13x^2 + 4
\][/tex]
2. Factor the quadratic form: Notice that this is a quadratic in terms of [tex]\(x^2\)[/tex]. Let's rewrite it as:
[tex]\[
(9x^2 - 4)(x^2 - 1)
\][/tex]
3. Set each factor equal to zero:
- For [tex]\(9x^2 - 4 = 0\)[/tex]:
[tex]\[
9x^2 = 4
\][/tex]
[tex]\[
x^2 = \frac{4}{9}
\][/tex]
[tex]\[
x = \pm \frac{2}{3}
\][/tex]
- For [tex]\(x^2 - 1 = 0\)[/tex]:
[tex]\[
x^2 = 1
\][/tex]
[tex]\[
x = \pm 1
\][/tex]
4. List all zeros: The zeros of the polynomial are [tex]\(x = -1, -\frac{2}{3}, \frac{2}{3}, 1\)[/tex].
Hence, the zeros of the polynomial [tex]\(9x^4 - 9x^2 - 4x^2 + 4\)[/tex] are [tex]\(-1\)[/tex], [tex]\(-\frac{2}{3}\)[/tex], [tex]\(\frac{2}{3}\)[/tex], and [tex]\(1\)[/tex].
1. Simplify: Combine like terms in the polynomial:
[tex]\[
9x^4 - (9x^2 + 4x^2) + 4 = 9x^4 - 13x^2 + 4
\][/tex]
2. Factor the quadratic form: Notice that this is a quadratic in terms of [tex]\(x^2\)[/tex]. Let's rewrite it as:
[tex]\[
(9x^2 - 4)(x^2 - 1)
\][/tex]
3. Set each factor equal to zero:
- For [tex]\(9x^2 - 4 = 0\)[/tex]:
[tex]\[
9x^2 = 4
\][/tex]
[tex]\[
x^2 = \frac{4}{9}
\][/tex]
[tex]\[
x = \pm \frac{2}{3}
\][/tex]
- For [tex]\(x^2 - 1 = 0\)[/tex]:
[tex]\[
x^2 = 1
\][/tex]
[tex]\[
x = \pm 1
\][/tex]
4. List all zeros: The zeros of the polynomial are [tex]\(x = -1, -\frac{2}{3}, \frac{2}{3}, 1\)[/tex].
Hence, the zeros of the polynomial [tex]\(9x^4 - 9x^2 - 4x^2 + 4\)[/tex] are [tex]\(-1\)[/tex], [tex]\(-\frac{2}{3}\)[/tex], [tex]\(\frac{2}{3}\)[/tex], and [tex]\(1\)[/tex].
