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Answer :
To factor the expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] completely, let's go through the process step-by-step:
1. Identify the Structure:
The expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] is a polynomial in terms of [tex]\( x^2 \)[/tex]. You can think of it in the form of a quadratic expression where [tex]\( y = x^2 \)[/tex]. Thus, we have [tex]\( y^2 - 4y - 21 \)[/tex].
2. Factor the Quadratic:
Now, we need to factor the quadratic expression [tex]\( y^2 - 4y - 21 \)[/tex]. We are looking for two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(-4\)[/tex].
The numbers that work are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex], since:
[tex]\[
-7 \times 3 = -21 \quad \text{and} \quad -7 + 3 = -4
\][/tex]
3. Rewrite the Expression:
Using these numbers, you can factor the quadratic as:
[tex]\[
y^2 - 4y - 21 = (y - 7)(y + 3)
\][/tex]
4. Substitute Back [tex]\( x^2 \)[/tex]:
Remember that [tex]\( y = x^2 \)[/tex], so substitute back to get:
[tex]\[
(x^2 - 7)(x^2 + 3)
\][/tex]
5. Conclusion:
The fully factored form of the original expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] is:
[tex]\[
(x^2 - 7)(x^2 + 3)
\][/tex]
Therefore, the expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] factors completely into [tex]\((x^2 - 7)(x^2 + 3)\)[/tex].
1. Identify the Structure:
The expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] is a polynomial in terms of [tex]\( x^2 \)[/tex]. You can think of it in the form of a quadratic expression where [tex]\( y = x^2 \)[/tex]. Thus, we have [tex]\( y^2 - 4y - 21 \)[/tex].
2. Factor the Quadratic:
Now, we need to factor the quadratic expression [tex]\( y^2 - 4y - 21 \)[/tex]. We are looking for two numbers that multiply to [tex]\(-21\)[/tex] and add up to [tex]\(-4\)[/tex].
The numbers that work are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex], since:
[tex]\[
-7 \times 3 = -21 \quad \text{and} \quad -7 + 3 = -4
\][/tex]
3. Rewrite the Expression:
Using these numbers, you can factor the quadratic as:
[tex]\[
y^2 - 4y - 21 = (y - 7)(y + 3)
\][/tex]
4. Substitute Back [tex]\( x^2 \)[/tex]:
Remember that [tex]\( y = x^2 \)[/tex], so substitute back to get:
[tex]\[
(x^2 - 7)(x^2 + 3)
\][/tex]
5. Conclusion:
The fully factored form of the original expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] is:
[tex]\[
(x^2 - 7)(x^2 + 3)
\][/tex]
Therefore, the expression [tex]\( x^4 - 4x^2 - 21 \)[/tex] factors completely into [tex]\((x^2 - 7)(x^2 + 3)\)[/tex].
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