Answer :
We want to factor the polynomial
[tex]$$24x^3 - 18x^2 - 60x + 45$$[/tex]
completely. Follow these steps:
1. Group the terms:
Group the polynomial into two pairs:
[tex]$$
(24x^3 - 18x^2) + (-60x + 45)
$$[/tex]
2. Factor each group:
- In the first group, [tex]\(24x^3 - 18x^2\)[/tex], the common factor is [tex]\(6x^2\)[/tex]:
[tex]$$
24x^3 - 18x^2 = 6x^2(4x - 3)
$$[/tex]
- In the second group, [tex]\(-60x + 45\)[/tex], the common factor is [tex]\(-15\)[/tex]:
[tex]$$
-60x + 45 = -15(4x - 3)
$$[/tex]
3. Factor out the common binomial:
Both groups now contain the factor [tex]\((4x-3)\)[/tex]. Factor [tex]\((4x-3)\)[/tex] from the entire expression:
[tex]$$
6x^2(4x-3) - 15(4x-3) = (4x-3)(6x^2 - 15)
$$[/tex]
4. Factor the quadratic expression:
Look at [tex]\(6x^2 - 15\)[/tex]. Notice that 3 is a common factor:
[tex]$$
6x^2 - 15 = 3(2x^2 - 5)
$$[/tex]
5. Write the complete factorization:
Substitute the factored form back into the expression:
[tex]$$
(4x-3)(6x^2 - 15) = (4x-3) \cdot 3(2x^2-5)
$$[/tex]
Rearranging slightly, the complete factorization is:
[tex]$$
3(4x-3)(2x^2-5)
$$[/tex]
Thus, the fully factored form of the polynomial is
[tex]$$\boxed{3(4x-3)(2x^2-5)}.$$[/tex]
[tex]$$24x^3 - 18x^2 - 60x + 45$$[/tex]
completely. Follow these steps:
1. Group the terms:
Group the polynomial into two pairs:
[tex]$$
(24x^3 - 18x^2) + (-60x + 45)
$$[/tex]
2. Factor each group:
- In the first group, [tex]\(24x^3 - 18x^2\)[/tex], the common factor is [tex]\(6x^2\)[/tex]:
[tex]$$
24x^3 - 18x^2 = 6x^2(4x - 3)
$$[/tex]
- In the second group, [tex]\(-60x + 45\)[/tex], the common factor is [tex]\(-15\)[/tex]:
[tex]$$
-60x + 45 = -15(4x - 3)
$$[/tex]
3. Factor out the common binomial:
Both groups now contain the factor [tex]\((4x-3)\)[/tex]. Factor [tex]\((4x-3)\)[/tex] from the entire expression:
[tex]$$
6x^2(4x-3) - 15(4x-3) = (4x-3)(6x^2 - 15)
$$[/tex]
4. Factor the quadratic expression:
Look at [tex]\(6x^2 - 15\)[/tex]. Notice that 3 is a common factor:
[tex]$$
6x^2 - 15 = 3(2x^2 - 5)
$$[/tex]
5. Write the complete factorization:
Substitute the factored form back into the expression:
[tex]$$
(4x-3)(6x^2 - 15) = (4x-3) \cdot 3(2x^2-5)
$$[/tex]
Rearranging slightly, the complete factorization is:
[tex]$$
3(4x-3)(2x^2-5)
$$[/tex]
Thus, the fully factored form of the polynomial is
[tex]$$\boxed{3(4x-3)(2x^2-5)}.$$[/tex]