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Answer :
Sure! Let's factor the polynomial [tex]\( x^4 - 4x^2 - 21 \)[/tex] completely. Here’s how you can approach it step-by-step:
1. Identify the form of the polynomial:
The polynomial is [tex]\( x^4 - 4x^2 - 21 \)[/tex]. Notice that this can be seen as a quadratic in terms of [tex]\( x^2 \)[/tex].
2. Substitution:
Let [tex]\( y = x^2 \)[/tex]. So the polynomial can be rewritten as:
[tex]\[ y^2 - 4y - 21 \][/tex]
3. Factor the quadratic:
We now want to factor [tex]\( y^2 - 4y - 21 \)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-21\)[/tex] and add to [tex]\(-4\)[/tex].
These numbers are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex]. Therefore, we can factor the expression as:
[tex]\[ y^2 - 4y - 21 = (y - 7)(y + 3) \][/tex]
4. Substitute back [tex]\( x^2 \)[/tex] for [tex]\( y \)[/tex]:
Recall [tex]\( y = x^2 \)[/tex]. Substitute back to get:
[tex]\[ (x^2 - 7)(x^2 + 3) \][/tex]
5. Final Factored Form:
The polynomial [tex]\( x^4 - 4x^2 - 21 \)[/tex] is factored completely as:
[tex]\[ (x^2 - 7)(x^2 + 3) \][/tex]
And there you have it, the complete factorization of the polynomial!
1. Identify the form of the polynomial:
The polynomial is [tex]\( x^4 - 4x^2 - 21 \)[/tex]. Notice that this can be seen as a quadratic in terms of [tex]\( x^2 \)[/tex].
2. Substitution:
Let [tex]\( y = x^2 \)[/tex]. So the polynomial can be rewritten as:
[tex]\[ y^2 - 4y - 21 \][/tex]
3. Factor the quadratic:
We now want to factor [tex]\( y^2 - 4y - 21 \)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-21\)[/tex] and add to [tex]\(-4\)[/tex].
These numbers are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex]. Therefore, we can factor the expression as:
[tex]\[ y^2 - 4y - 21 = (y - 7)(y + 3) \][/tex]
4. Substitute back [tex]\( x^2 \)[/tex] for [tex]\( y \)[/tex]:
Recall [tex]\( y = x^2 \)[/tex]. Substitute back to get:
[tex]\[ (x^2 - 7)(x^2 + 3) \][/tex]
5. Final Factored Form:
The polynomial [tex]\( x^4 - 4x^2 - 21 \)[/tex] is factored completely as:
[tex]\[ (x^2 - 7)(x^2 + 3) \][/tex]
And there you have it, the complete factorization of the polynomial!
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