Answer :

Sure! Let's factor the polynomial [tex]\( 6x^5 - 21x^4 + 15x^3 \)[/tex] completely.

### Step 1: Factor out the Greatest Common Factor (GCF)
First, we notice that each term in the polynomial shares some common factors. Specifically, all terms contain [tex]\( x^3 \)[/tex], and the numerical coefficients (6, 21, and 15) have a greatest common factor of 3. Therefore, we can factor out [tex]\( 3x^3 \)[/tex] from each term:

[tex]\[
6x^5 - 21x^4 + 15x^3 = 3x^3 (2x^2) - 3x^3 (7x) + 3x^3 (5)
\][/tex]

This simplifies to:

[tex]\[
6x^5 - 21x^4 + 15x^3 = 3x^3 (2x^2 - 7x + 5)
\][/tex]

### Step 2: Factor the quadratic [tex]\( 2x^2 - 7x + 5 \)[/tex]

Now we need to factor the quadratic factor [tex]\( 2x^2 - 7x + 5 \)[/tex]. We look for two numbers that multiply to [tex]\( 2 \cdot 5 = 10 \)[/tex] (the product of the coefficient of [tex]\( x^2 \)[/tex] and the constant term) and add up to [tex]\(-7\)[/tex] (the coefficient of [tex]\( x \)[/tex]).

After some inspection, we find the numbers are [tex]\(-2\)[/tex] and [tex]\(-5\)[/tex]:

[tex]\[
2x^2 - 7x + 5 = 2x^2 - 2x - 5x + 5
\][/tex]

### Step 3: Factor by grouping

Group terms to factor by grouping:

[tex]\[
2x^2 - 2x - 5x + 5 = 2x(x - 1) - 5(x - 1)
\][/tex]

Notice that [tex]\((x - 1)\)[/tex] is a common factor:

[tex]\[
2x(x - 1) - 5(x - 1) = (2x - 5)(x - 1)
\][/tex]

### Step 4: Combine the results

Putting it all together, the completely factored form of [tex]\( 6x^5 - 21x^4 + 15x^3 \)[/tex] is:

[tex]\[
6x^5 - 21x^4 + 15x^3 = 3x^3 (2x^2 - 7x + 5) = 3x^3 (2x - 5)(x - 1)
\][/tex]

So the final factored form is:

[tex]\[
\boxed{3x^3 (2x - 5)(x - 1)}
\][/tex]

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