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Answer :
To factor the polynomial [tex]\(2x^9 - 4x^5 - 16x\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, observe all the terms in the polynomial. Notice that each term has a common factor of [tex]\(2x\)[/tex]. So, let's factor out the GCF:
[tex]\[
2x(x^8 - 2x^4 - 8)
\][/tex]
2. Factor the Remaining Polynomial:
Next, we need to factor the polynomial inside the parentheses, [tex]\(x^8 - 2x^4 - 8\)[/tex]. This polynomial can be seen as a quadratic in terms of [tex]\(x^4\)[/tex].
3. Substitution Approach:
Let's use a substitution to make it easier to factor. Set [tex]\(u = x^4\)[/tex]. Then the polynomial becomes:
[tex]\[
u^2 - 2u - 8
\][/tex]
4. Factor the Quadratic Expression:
Now, factor the expression [tex]\(u^2 - 2u - 8\)[/tex]. We need to find two numbers that multiply to [tex]\(-8\)[/tex] and add to [tex]\(-2\)[/tex]. These numbers are [tex]\(-4\)[/tex] and [tex]\(2\)[/tex]. Therefore, the factorization is:
[tex]\[
(u - 4)(u + 2)
\][/tex]
5. Substitute Back:
Replace [tex]\(u\)[/tex] with [tex]\(x^4\)[/tex] in the factors:
[tex]\[
(x^4 - 4)(x^4 + 2)
\][/tex]
6. Further Factorization:
Notice that [tex]\(x^4 - 4\)[/tex] can be factored as a difference of squares:
[tex]\[
x^4 - 4 = (x^2 - 2)(x^2 + 2)
\][/tex]
7. Combine All Factors:
Now, combine all the factors we have found, including the GCF we factored out at the beginning:
[tex]\[
2x(x^2 - 2)(x^2 + 2)(x^4 + 2)
\][/tex]
This is the complete factorization of the polynomial [tex]\(2x^9 - 4x^5 - 16x\)[/tex].
1. Identify the Greatest Common Factor (GCF):
First, observe all the terms in the polynomial. Notice that each term has a common factor of [tex]\(2x\)[/tex]. So, let's factor out the GCF:
[tex]\[
2x(x^8 - 2x^4 - 8)
\][/tex]
2. Factor the Remaining Polynomial:
Next, we need to factor the polynomial inside the parentheses, [tex]\(x^8 - 2x^4 - 8\)[/tex]. This polynomial can be seen as a quadratic in terms of [tex]\(x^4\)[/tex].
3. Substitution Approach:
Let's use a substitution to make it easier to factor. Set [tex]\(u = x^4\)[/tex]. Then the polynomial becomes:
[tex]\[
u^2 - 2u - 8
\][/tex]
4. Factor the Quadratic Expression:
Now, factor the expression [tex]\(u^2 - 2u - 8\)[/tex]. We need to find two numbers that multiply to [tex]\(-8\)[/tex] and add to [tex]\(-2\)[/tex]. These numbers are [tex]\(-4\)[/tex] and [tex]\(2\)[/tex]. Therefore, the factorization is:
[tex]\[
(u - 4)(u + 2)
\][/tex]
5. Substitute Back:
Replace [tex]\(u\)[/tex] with [tex]\(x^4\)[/tex] in the factors:
[tex]\[
(x^4 - 4)(x^4 + 2)
\][/tex]
6. Further Factorization:
Notice that [tex]\(x^4 - 4\)[/tex] can be factored as a difference of squares:
[tex]\[
x^4 - 4 = (x^2 - 2)(x^2 + 2)
\][/tex]
7. Combine All Factors:
Now, combine all the factors we have found, including the GCF we factored out at the beginning:
[tex]\[
2x(x^2 - 2)(x^2 + 2)(x^4 + 2)
\][/tex]
This is the complete factorization of the polynomial [tex]\(2x^9 - 4x^5 - 16x\)[/tex].
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