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Answer :
To factor the expression [tex]\(24x^{14} + 24x^9 + 4x^5\)[/tex] completely, we can follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, we look for the greatest common factor of all the terms in the expression. The coefficients (24, 24, and 4) have a GCF of 4. Next, looking at the powers of [tex]\(x\)[/tex] (14, 9, and 5), the smallest power is 5. Therefore, the GCF of the expression is [tex]\(4x^5\)[/tex].
2. Factor out the GCF:
We can factor [tex]\(4x^5\)[/tex] out of each term in the expression:
[tex]\[
24x^{14} + 24x^9 + 4x^5 = 4x^5(6x^9 + 6x^4 + 1)
\][/tex]
3. Check the Factored Expression:
The expression inside the parentheses, [tex]\(6x^9 + 6x^4 + 1\)[/tex], should be examined to see if it can be factored further. However, this trinomial does not have any further common factors and does not fit the patterns for special products or further factorization using simple integer coefficients.
Therefore, the fully factored form of the expression [tex]\(24x^{14} + 24x^9 + 4x^5\)[/tex] is:
[tex]\[
4x^5(6x^9 + 6x^4 + 1)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
First, we look for the greatest common factor of all the terms in the expression. The coefficients (24, 24, and 4) have a GCF of 4. Next, looking at the powers of [tex]\(x\)[/tex] (14, 9, and 5), the smallest power is 5. Therefore, the GCF of the expression is [tex]\(4x^5\)[/tex].
2. Factor out the GCF:
We can factor [tex]\(4x^5\)[/tex] out of each term in the expression:
[tex]\[
24x^{14} + 24x^9 + 4x^5 = 4x^5(6x^9 + 6x^4 + 1)
\][/tex]
3. Check the Factored Expression:
The expression inside the parentheses, [tex]\(6x^9 + 6x^4 + 1\)[/tex], should be examined to see if it can be factored further. However, this trinomial does not have any further common factors and does not fit the patterns for special products or further factorization using simple integer coefficients.
Therefore, the fully factored form of the expression [tex]\(24x^{14} + 24x^9 + 4x^5\)[/tex] is:
[tex]\[
4x^5(6x^9 + 6x^4 + 1)
\][/tex]
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