Answer :

Let's factor the polynomial [tex]\(9x^5 - 33x^4 - 60x^3\)[/tex] completely, step by step.

1. Identify the Greatest Common Factor (GCF):

First, we look for the greatest common factor of all the terms in the polynomial:

- All terms have a factor of [tex]\(3x^3\)[/tex].

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

2. Factor out the GCF:

When we factor [tex]\(3x^3\)[/tex] out of each term, our polynomial changes as follows:

[tex]\[
9x^5 \div 3x^3 = 3x^2
\][/tex]

[tex]\[
-33x^4 \div 3x^3 = -11x
\][/tex]

[tex]\[
-60x^3 \div 3x^3 = -20
\][/tex]

This gives us:

[tex]\[
3x^3 (3x^2 - 11x - 20)
\][/tex]

3. Factor the quadratic expression inside the parentheses:

Now we need to factor the quadratic [tex]\(3x^2 - 11x - 20\)[/tex].

To factor this, we look for two numbers that multiply to [tex]\((3 \times -20 = -60)\)[/tex] and add to [tex]\(-11\)[/tex]. The numbers [tex]\(-15\)[/tex] and [tex]\(4\)[/tex] do the job because:

[tex]\(-15 \times 4 = -60\)[/tex]

[tex]\(-15 + 4 = -11\)[/tex]

We can then write the middle term, [tex]\(-11x\)[/tex], as [tex]\(-15x + 4x\)[/tex]:

[tex]\[
3x^2 - 15x + 4x - 20
\][/tex]

Group the terms to factor by grouping:

[tex]\[
= (3x^2 - 15x) + (4x - 20)
\][/tex]

Factor out the GCF from each group:

[tex]\[
= 3x(x - 5) + 4(x - 5)
\][/tex]

Since both terms contain a common factor of [tex]\((x - 5)\)[/tex], we can factor it out:

[tex]\[
= (x - 5)(3x + 4)
\][/tex]

4. Write the completely factored form:

With everything combined, the completely factored form of the polynomial is:

[tex]\[
3x^3(x - 5)(3x + 4)
\][/tex]

And that's the complete factorization of the polynomial [tex]\(9x^5 - 33x^4 - 60x^3\)[/tex].

Other Questions