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Answer :
To factor the polynomial [tex]\( 7x^7 - 35x^6 + 21x^5 \)[/tex], follow these steps:
1. Identify the Greatest Common Factor (GCF):
- First, look at the coefficients: 7, 35, and 21. The greatest common factor here is 7.
- Next, examine the variable parts: [tex]\( x^7 \)[/tex], [tex]\( x^6 \)[/tex], and [tex]\( x^5 \)[/tex]. The smallest power of [tex]\( x \)[/tex] is [tex]\( x^5 \)[/tex].
So, the GCF of the entire polynomial is [tex]\( 7x^5 \)[/tex].
2. Factor Out the GCF:
- Divide each term of the polynomial by the GCF [tex]\( 7x^5 \)[/tex].
- This results in:
[tex]\[
\frac{7x^7}{7x^5} = x^2, \quad \frac{-35x^6}{7x^5} = -5x, \quad \frac{21x^5}{7x^5} = 3
\][/tex]
- Write the polynomial as a product of the GCF and the resulting expression:
[tex]\[
7x^5(x^2 - 5x + 3)
\][/tex]
3. Check for Further Factoring:
- Now, look at the quadratic part [tex]\( x^2 - 5x + 3 \)[/tex].
- Since it doesn't factor further over the integers, the factored form of the polynomial remains:
[tex]\[
7x^5(x^2 - 5x + 3)
\][/tex]
Thus, the polynomial [tex]\( 7x^7 - 35x^6 + 21x^5 \)[/tex] factors to [tex]\( 7x^5(x^2 - 5x + 3) \)[/tex].
1. Identify the Greatest Common Factor (GCF):
- First, look at the coefficients: 7, 35, and 21. The greatest common factor here is 7.
- Next, examine the variable parts: [tex]\( x^7 \)[/tex], [tex]\( x^6 \)[/tex], and [tex]\( x^5 \)[/tex]. The smallest power of [tex]\( x \)[/tex] is [tex]\( x^5 \)[/tex].
So, the GCF of the entire polynomial is [tex]\( 7x^5 \)[/tex].
2. Factor Out the GCF:
- Divide each term of the polynomial by the GCF [tex]\( 7x^5 \)[/tex].
- This results in:
[tex]\[
\frac{7x^7}{7x^5} = x^2, \quad \frac{-35x^6}{7x^5} = -5x, \quad \frac{21x^5}{7x^5} = 3
\][/tex]
- Write the polynomial as a product of the GCF and the resulting expression:
[tex]\[
7x^5(x^2 - 5x + 3)
\][/tex]
3. Check for Further Factoring:
- Now, look at the quadratic part [tex]\( x^2 - 5x + 3 \)[/tex].
- Since it doesn't factor further over the integers, the factored form of the polynomial remains:
[tex]\[
7x^5(x^2 - 5x + 3)
\][/tex]
Thus, the polynomial [tex]\( 7x^7 - 35x^6 + 21x^5 \)[/tex] factors to [tex]\( 7x^5(x^2 - 5x + 3) \)[/tex].
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