Answer :

Sure! Let's factor out the greatest common factor from the polynomial [tex]\(7x^7 - 28x^6 + 35x^5\)[/tex].

Step 1: Identify the coefficients.

The given polynomial is:

[tex]\[ 7x^7 - 28x^6 + 35x^5 \][/tex]

The coefficients of the polynomial are 7, -28, and 35.

Step 2: Find the greatest common factor (GCF) of the coefficients.

The numbers we are working with are 7, 28, and 35. Let's find the GCF of these numbers:

- The prime factorization of 7 is [tex]\(7\)[/tex].
- The prime factorization of 28 is [tex]\(2^2 \times 7\)[/tex].
- The prime factorization of 35 is [tex]\(5 \times 7\)[/tex].

The only common prime factor is [tex]\(7\)[/tex].

So, the GCF of the coefficients is 7.

Step 3: Factor the GCF out of the entire expression.

To factor the GCF out of each term, we divide each coefficient by 7:

- The first term: [tex]\( \frac{7x^7}{7} = x^7 \)[/tex]
- The second term: [tex]\( \frac{-28x^6}{7} = -4x^6 \)[/tex]
- The third term: [tex]\( \frac{35x^5}{7} = 5x^5 \)[/tex]

Step 4: Write the factored expression.

Now, we can express the original polynomial as:

[tex]\[ 7(x^7 - 4x^6 + 5x^5) \][/tex]

This is the polynomial factored by the greatest common factor.

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