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Answer :
To simplify the expression [tex]\(\frac{x^4 - 81}{x^2 - 9}\)[/tex], let's go through it step-by-step:
1. Recognize the type of expressions:
- The numerator, [tex]\(x^4 - 81\)[/tex], is a difference of squares. It can be rewritten as [tex]\((x^2)^2 - 9^2\)[/tex].
- The denominator, [tex]\(x^2 - 9\)[/tex], is also a difference of squares. It can be rewritten as [tex]\((x)^2 - (3)^2\)[/tex].
2. Factor the numerator and the denominator:
- For the numerator, [tex]\(x^4 - 81\)[/tex], we have:
[tex]\[
x^4 - 81 = (x^2 - 9)(x^2 + 9)
\][/tex]
The factor [tex]\(x^2 - 9\)[/tex] is another difference of squares and can be further factored:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
So, the factored form of the numerator is:
[tex]\[
(x - 3)(x + 3)(x^2 + 9)
\][/tex]
- For the denominator, [tex]\(x^2 - 9\)[/tex]:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Simplify the expression:
- Now, substitute the factored forms into the original expression:
[tex]\[
\frac{(x - 3)(x + 3)(x^2 + 9)}{(x - 3)(x + 3)}
\][/tex]
- Cancel out the common factors [tex]\((x - 3)\)[/tex] and [tex]\((x + 3)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{\cancel{(x - 3)}\cancel{(x + 3)}(x^2 + 9)}{\cancel{(x - 3)}\cancel{(x + 3)}} = x^2 + 9
\][/tex]
4. Result:
The simplified expression is [tex]\(x^2 + 9\)[/tex].
The correct answer is A) [tex]\(x^2 + 9\)[/tex].
1. Recognize the type of expressions:
- The numerator, [tex]\(x^4 - 81\)[/tex], is a difference of squares. It can be rewritten as [tex]\((x^2)^2 - 9^2\)[/tex].
- The denominator, [tex]\(x^2 - 9\)[/tex], is also a difference of squares. It can be rewritten as [tex]\((x)^2 - (3)^2\)[/tex].
2. Factor the numerator and the denominator:
- For the numerator, [tex]\(x^4 - 81\)[/tex], we have:
[tex]\[
x^4 - 81 = (x^2 - 9)(x^2 + 9)
\][/tex]
The factor [tex]\(x^2 - 9\)[/tex] is another difference of squares and can be further factored:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
So, the factored form of the numerator is:
[tex]\[
(x - 3)(x + 3)(x^2 + 9)
\][/tex]
- For the denominator, [tex]\(x^2 - 9\)[/tex]:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Simplify the expression:
- Now, substitute the factored forms into the original expression:
[tex]\[
\frac{(x - 3)(x + 3)(x^2 + 9)}{(x - 3)(x + 3)}
\][/tex]
- Cancel out the common factors [tex]\((x - 3)\)[/tex] and [tex]\((x + 3)\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{\cancel{(x - 3)}\cancel{(x + 3)}(x^2 + 9)}{\cancel{(x - 3)}\cancel{(x + 3)}} = x^2 + 9
\][/tex]
4. Result:
The simplified expression is [tex]\(x^2 + 9\)[/tex].
The correct answer is A) [tex]\(x^2 + 9\)[/tex].
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