Answer :

Certainly! Let's factor the expression [tex]\(7x^5 - 35x^3\)[/tex] step-by-step and then check our work by multiplying to verify.

### Step 1: Find the Greatest Common Factor (GCF)
First, we need to determine the greatest common factor of the two terms in the expression.

- Coefficients: The coefficients are 7 and 35. The greatest common divisor of 7 and 35 is 7.
- Variables and Exponents: The terms are [tex]\(x^5\)[/tex] and [tex]\(x^3\)[/tex]. The smallest power of [tex]\(x\)[/tex] common to both is [tex]\(x^3\)[/tex].

Combining these, the GCF of the entire expression is [tex]\(7x^3\)[/tex].

### Step 2: Factor Out the GCF
Now we factor out [tex]\(7x^3\)[/tex] from each term in the expression:

- Divide the first term [tex]\(7x^5\)[/tex] by [tex]\(7x^3\)[/tex]:
[tex]\[
\frac{7x^5}{7x^3} = x^2
\][/tex]
- Divide the second term [tex]\(-35x^3\)[/tex] by [tex]\(7x^3\)[/tex]:
[tex]\[
\frac{-35x^3}{7x^3} = -5
\][/tex]

Putting it all together, the factored expression is:
[tex]\[
7x^3(x^2 - 5)
\][/tex]

### Step 3: Verify by Multiplying
To ensure our factorization is correct, let's multiply the factors and see if we arrive back at the original expression:

- Multiply [tex]\(7x^3\)[/tex] by each term inside the parentheses:
- [tex]\(7x^3 \cdot x^2 = 7x^5\)[/tex]
- [tex]\(7x^3 \cdot (-5) = -35x^3\)[/tex]

Combining these results, we get back the original expression:
[tex]\[
7x^5 - 35x^3
\][/tex]

Since our multiplication gives us the original expression, the factorization is confirmed to be correct.

Therefore, the expression [tex]\(7x^5 - 35x^3\)[/tex] factors to [tex]\(7x^3(x^2 - 5)\)[/tex].

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Rewritten by : Barada