Let's determine the completely factored form of the polynomial [tex]\( x^4 + 8x^2 - 9 \)[/tex].
Given the polynomial:
[tex]\[ x^4 + 8x^2 - 9 \][/tex]
We aim to break it down into simpler factors. Here’s the step-by-step process:
1. Analyze the polynomial:
The polynomial is a quadratic in form, [tex]\( x^4 + 8x^2 - 9 \)[/tex], where we can treat [tex]\( x^2 \)[/tex] as a single variable.
2. Rewrite the polynomial:
Rewrite the expression by introducing a new variable, let [tex]\( y = x^2 \)[/tex]. Then, our polynomial becomes:
[tex]\[ y^2 + 8y - 9 \][/tex]
3. Factor the quadratic:
We now factor [tex]\( y^2 + 8y - 9 \)[/tex]. When factoring a quadratic [tex]\( ay^2 + by + c \)[/tex], we look for two numbers that multiply to [tex]\( c \)[/tex] (-9) and add up to [tex]\( b \)[/tex] (8).
Here, we have:
[tex]\[ y^2 + 8y - 9 = (y + 9)(y - 1) \][/tex]
4. Re-substitute back:
We substitute back [tex]\( y = x^2 \)[/tex] into our factors:
[tex]\[ (x^2 + 9)(x^2 - 1) \][/tex]
5. Further factorize:
Notice that [tex]\( x^2 - 1 \)[/tex] is a difference of squares and can be further factored:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
Thus, our expression becomes:
[tex]\[ (x^2 + 9)(x - 1)(x + 1) \][/tex]
The completely factored form of [tex]\( x^4 + 8x^2 - 9 \)[/tex] is:
[tex]\[ (x - 1)(x + 1)(x^2 + 9) \][/tex]
So the correct answer from the given choices is:
[tex]\[ (x+1)(x-1)(x^2+9) \][/tex]
