Answer :

Sure! Let's break down the steps to factor the expression [tex]\(21x^7 + 9x^6 + 33x^4\)[/tex] completely.

### Step 1: Identify the Greatest Common Factor (GCF)
First, we look for the greatest common factor (GCF) among the terms. The terms are [tex]\(21x^7\)[/tex], [tex]\(9x^6\)[/tex], and [tex]\(33x^4\)[/tex].

- For the coefficients 21, 9, and 33, the GCF is 3.
- For the variable part [tex]\(x^7\)[/tex], [tex]\(x^6\)[/tex], and [tex]\(x^4\)[/tex], the GCF is [tex]\(x^4\)[/tex] (the lowest power of [tex]\(x\)[/tex] common to all terms).

So, the GCF of the entire expression is [tex]\(3x^4\)[/tex].

### Step 2: Factor out the GCF
We factor out the GCF [tex]\(3x^4\)[/tex] from each term in the expression:
[tex]\[
21x^7 + 9x^6 + 33x^4 = 3x^4(7x^3) + 3x^4(3x^2) + 3x^4(11).
\][/tex]

This simplifies to:
[tex]\[
21x^7 + 9x^6 + 33x^4 = 3x^4(7x^3 + 3x^2 + 11).
\][/tex]

### Step 3: Verify the factored expression
We check that if we distribute [tex]\(3x^4\)[/tex] back into the terms within the parentheses, we get the original expression:
[tex]\[
3x^4 \cdot 7x^3 + 3x^4 \cdot 3x^2 + 3x^4 \cdot 11 = 21x^7 + 9x^6 + 33x^4.
\][/tex]

Since the factorization holds true, the completely factored form of the expression is:
[tex]\[
\boxed{3x^4(7x^3 + 3x^2 + 11)}
\][/tex]

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