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Answer :
To factor the trinomial [tex]\(5x^7 + 60x^6 + 160x^5\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 5, 60, and 160. The GCF of these numbers is 5.
- Look at the variable part: [tex]\(x^7\)[/tex], [tex]\(x^6\)[/tex], and [tex]\(x^5\)[/tex]. The least power of [tex]\(x\)[/tex] present in all terms is [tex]\(x^5\)[/tex].
- Therefore, the GCF of the entire trinomial is [tex]\(5x^5\)[/tex].
2. Factor out the GCF:
- Divide each term in the trinomial by the GCF [tex]\(5x^5\)[/tex]:
- [tex]\(5x^7 \div 5x^5 = x^2\)[/tex]
- [tex]\(60x^6 \div 5x^5 = 12x\)[/tex]
- [tex]\(160x^5 \div 5x^5 = 32\)[/tex]
- So, factoring out the GCF gives us:
[tex]\[
5x^7 + 60x^6 + 160x^5 = 5x^5(x^2 + 12x + 32)
\][/tex]
3. Factor the Remaining Quadratic (if possible):
- Look at the quadratic [tex]\(x^2 + 12x + 32\)[/tex]. Check if it can be factored further.
- We need to find two numbers that multiply to 32 and add to 12. These numbers are 4 and 8.
- So, the quadratic can be factored as:
[tex]\[
x^2 + 12x + 32 = (x + 4)(x + 8)
\][/tex]
4. Write the Fully Factored Form:
- Combine the factored parts to express the completely factored form of the original trinomial:
[tex]\[
5x^7 + 60x^6 + 160x^5 = 5x^5(x + 4)(x + 8)
\][/tex]
Thus, the correct choice is:
A. [tex]\(5x^7+60x^6+160x^5 = 5x^5(x + 4)(x + 8)\)[/tex]
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 5, 60, and 160. The GCF of these numbers is 5.
- Look at the variable part: [tex]\(x^7\)[/tex], [tex]\(x^6\)[/tex], and [tex]\(x^5\)[/tex]. The least power of [tex]\(x\)[/tex] present in all terms is [tex]\(x^5\)[/tex].
- Therefore, the GCF of the entire trinomial is [tex]\(5x^5\)[/tex].
2. Factor out the GCF:
- Divide each term in the trinomial by the GCF [tex]\(5x^5\)[/tex]:
- [tex]\(5x^7 \div 5x^5 = x^2\)[/tex]
- [tex]\(60x^6 \div 5x^5 = 12x\)[/tex]
- [tex]\(160x^5 \div 5x^5 = 32\)[/tex]
- So, factoring out the GCF gives us:
[tex]\[
5x^7 + 60x^6 + 160x^5 = 5x^5(x^2 + 12x + 32)
\][/tex]
3. Factor the Remaining Quadratic (if possible):
- Look at the quadratic [tex]\(x^2 + 12x + 32\)[/tex]. Check if it can be factored further.
- We need to find two numbers that multiply to 32 and add to 12. These numbers are 4 and 8.
- So, the quadratic can be factored as:
[tex]\[
x^2 + 12x + 32 = (x + 4)(x + 8)
\][/tex]
4. Write the Fully Factored Form:
- Combine the factored parts to express the completely factored form of the original trinomial:
[tex]\[
5x^7 + 60x^6 + 160x^5 = 5x^5(x + 4)(x + 8)
\][/tex]
Thus, the correct choice is:
A. [tex]\(5x^7+60x^6+160x^5 = 5x^5(x + 4)(x + 8)\)[/tex]
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