Answer :
Sure! Let's factor the polynomial [tex]\( x^4 - 4x^2 - 21 \)[/tex] step by step.
1. Recognize the Structure:
Notice that the expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] resembles a quadratic expression, with [tex]\( x^4 \)[/tex] and [tex]\( x^2 \)[/tex] similar to [tex]\( (x^2)^2 \)[/tex] and [tex]\( x^2 \)[/tex], respectively. This suggests that we can treat it like a quadratic in terms of [tex]\( x^2 \)[/tex].
2. Substitute a New Variable:
Temporarily, let [tex]\( y = x^2 \)[/tex]. Then the expression becomes:
[tex]\[
y^2 - 4y - 21
\][/tex]
This looks like a standard quadratic equation which is easier to factor.
3. Factor the Quadratic:
We need to find two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(-4\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex].
4. Write the Factored Form:
Using these numbers, we can factor the quadratic as:
[tex]\[
(y - 7)(y + 3)
\][/tex]
5. Substitute Back [tex]\( x^2 \)[/tex] for [tex]\( y \)[/tex]:
Replace [tex]\( y \)[/tex] with [tex]\( x^2 \)[/tex] to come back to the original variable:
[tex]\[
(x^2 - 7)(x^2 + 3)
\][/tex]
6. Final Factored Form:
The polynomial [tex]\( x^4 - 4x^2 - 21 \)[/tex] is factored as [tex]\( (x^2 - 7)(x^2 + 3) \)[/tex].
This solution explains how to factor the polynomial by recognizing it as a quadratic in disguise, using substitution, and then factoring it using standard methods.
1. Recognize the Structure:
Notice that the expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] resembles a quadratic expression, with [tex]\( x^4 \)[/tex] and [tex]\( x^2 \)[/tex] similar to [tex]\( (x^2)^2 \)[/tex] and [tex]\( x^2 \)[/tex], respectively. This suggests that we can treat it like a quadratic in terms of [tex]\( x^2 \)[/tex].
2. Substitute a New Variable:
Temporarily, let [tex]\( y = x^2 \)[/tex]. Then the expression becomes:
[tex]\[
y^2 - 4y - 21
\][/tex]
This looks like a standard quadratic equation which is easier to factor.
3. Factor the Quadratic:
We need to find two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(-4\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex].
4. Write the Factored Form:
Using these numbers, we can factor the quadratic as:
[tex]\[
(y - 7)(y + 3)
\][/tex]
5. Substitute Back [tex]\( x^2 \)[/tex] for [tex]\( y \)[/tex]:
Replace [tex]\( y \)[/tex] with [tex]\( x^2 \)[/tex] to come back to the original variable:
[tex]\[
(x^2 - 7)(x^2 + 3)
\][/tex]
6. Final Factored Form:
The polynomial [tex]\( x^4 - 4x^2 - 21 \)[/tex] is factored as [tex]\( (x^2 - 7)(x^2 + 3) \)[/tex].
This solution explains how to factor the polynomial by recognizing it as a quadratic in disguise, using substitution, and then factoring it using standard methods.
