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Answer :
To solve the expression [tex]\(x^4 - 9x^2\)[/tex], we can factor it step by step. Here is how you can do it:
1. Identify the Greatest Common Factor (GCF):
The first step in factoring is to look for a GCF in the terms. In this expression, both terms [tex]\(x^4\)[/tex] and [tex]\(-9x^2\)[/tex] share a common factor of [tex]\(x^2\)[/tex].
2. Factor out the GCF:
When you factor [tex]\(x^2\)[/tex] out of each term, you get:
[tex]\[
x^4 - 9x^2 = x^2(x^2 - 9)
\][/tex]
3. Recognize a Difference of Squares:
The expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. Recall that the difference of squares formula is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(x^2\)[/tex] is [tex]\(a^2\)[/tex] and [tex]\(9\)[/tex] is [tex]\(b^2\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex].
4. Apply the Difference of Squares Formula:
Substitute [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex] into the formula:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
5. Combine the Factors:
Now, substitute back into the expression and write the fully factored form:
[tex]\[
x^2(x^2 - 9) = x^2(x - 3)(x + 3)
\][/tex]
So, the expression [tex]\(x^4 - 9x^2\)[/tex] factors to [tex]\(x^2(x - 3)(x + 3)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
The first step in factoring is to look for a GCF in the terms. In this expression, both terms [tex]\(x^4\)[/tex] and [tex]\(-9x^2\)[/tex] share a common factor of [tex]\(x^2\)[/tex].
2. Factor out the GCF:
When you factor [tex]\(x^2\)[/tex] out of each term, you get:
[tex]\[
x^4 - 9x^2 = x^2(x^2 - 9)
\][/tex]
3. Recognize a Difference of Squares:
The expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. Recall that the difference of squares formula is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(x^2\)[/tex] is [tex]\(a^2\)[/tex] and [tex]\(9\)[/tex] is [tex]\(b^2\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex].
4. Apply the Difference of Squares Formula:
Substitute [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex] into the formula:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
5. Combine the Factors:
Now, substitute back into the expression and write the fully factored form:
[tex]\[
x^2(x^2 - 9) = x^2(x - 3)(x + 3)
\][/tex]
So, the expression [tex]\(x^4 - 9x^2\)[/tex] factors to [tex]\(x^2(x - 3)(x + 3)\)[/tex].
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