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Answer :
To factor the polynomial [tex]\(7x^8 - 56x^7 + 28x^6\)[/tex], let's follow these steps:
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 7, 56, and 28. The greatest common factor of these numbers is 7.
- Look at the variable terms: [tex]\(x^8\)[/tex], [tex]\(x^7\)[/tex], and [tex]\(x^6\)[/tex]. The smallest power of [tex]\(x\)[/tex] is [tex]\(x^6\)[/tex].
- So, the GCF of the entire polynomial is [tex]\(7x^6\)[/tex].
2. Factor out the GCF:
- Divide each term in the polynomial by the GCF [tex]\(7x^6\)[/tex]:
- [tex]\(7x^8 \div 7x^6 = x^2\)[/tex]
- [tex]\(-56x^7 \div 7x^6 = -8x\)[/tex]
- [tex]\(28x^6 \div 7x^6 = 4\)[/tex]
3. Write the factored form:
- After factoring out the GCF, the expression becomes:
[tex]\[ 7x^6(x^2 - 8x + 4) \][/tex]
The factored form of the polynomial [tex]\(7x^8 - 56x^7 + 28x^6\)[/tex] is:
[tex]\[ 7x^6(x^2 - 8x + 4) \][/tex]
This gives us the complete factored form.
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 7, 56, and 28. The greatest common factor of these numbers is 7.
- Look at the variable terms: [tex]\(x^8\)[/tex], [tex]\(x^7\)[/tex], and [tex]\(x^6\)[/tex]. The smallest power of [tex]\(x\)[/tex] is [tex]\(x^6\)[/tex].
- So, the GCF of the entire polynomial is [tex]\(7x^6\)[/tex].
2. Factor out the GCF:
- Divide each term in the polynomial by the GCF [tex]\(7x^6\)[/tex]:
- [tex]\(7x^8 \div 7x^6 = x^2\)[/tex]
- [tex]\(-56x^7 \div 7x^6 = -8x\)[/tex]
- [tex]\(28x^6 \div 7x^6 = 4\)[/tex]
3. Write the factored form:
- After factoring out the GCF, the expression becomes:
[tex]\[ 7x^6(x^2 - 8x + 4) \][/tex]
The factored form of the polynomial [tex]\(7x^8 - 56x^7 + 28x^6\)[/tex] is:
[tex]\[ 7x^6(x^2 - 8x + 4) \][/tex]
This gives us the complete factored form.
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