Answer :
To factor the expression [tex]\(12x^3 - 19x^2 + 5x\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for a common factor in all the terms of the expression. Here, each term contains at least one [tex]\(x\)[/tex]. Therefore, the GCF is [tex]\(x\)[/tex].
Factor out [tex]\(x\)[/tex] from the expression:
[tex]\[
12x^3 - 19x^2 + 5x = x(12x^2 - 19x + 5)
\][/tex]
2. Factor the Quadratic Expression:
Now, focus on factoring the quadratic [tex]\(12x^2 - 19x + 5\)[/tex].
To factor the quadratic, look for two numbers that multiply to [tex]\(12 \times 5 = 60\)[/tex] and add up to [tex]\(-19\)[/tex].
The numbers that work are [tex]\(-15\)[/tex] and [tex]\(-4\)[/tex].
3. Rewrite and Factor by Grouping:
Rewrite [tex]\(-19x\)[/tex] as [tex]\(-15x - 4x\)[/tex] and then use factoring by grouping:
[tex]\[
12x^2 - 19x + 5 = 12x^2 - 15x - 4x + 5
\][/tex]
Group the terms:
[tex]\[
(12x^2 - 15x) + (-4x + 5)
\][/tex]
Factor out the GCF from each group:
[tex]\[
3x(4x - 5) - 1(4x - 5)
\][/tex]
Now factor out the common factor [tex]\((4x - 5)\)[/tex]:
[tex]\[
(3x - 1)(4x - 5)
\][/tex]
4. Combine the Factors:
Substitute back into the expression:
[tex]\[
x(12x^2 - 19x + 5) = x(3x - 1)(4x - 5)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(12x^3 - 19x^2 + 5x\)[/tex] is:
[tex]\[
x(3x - 1)(4x - 5)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
First, look for a common factor in all the terms of the expression. Here, each term contains at least one [tex]\(x\)[/tex]. Therefore, the GCF is [tex]\(x\)[/tex].
Factor out [tex]\(x\)[/tex] from the expression:
[tex]\[
12x^3 - 19x^2 + 5x = x(12x^2 - 19x + 5)
\][/tex]
2. Factor the Quadratic Expression:
Now, focus on factoring the quadratic [tex]\(12x^2 - 19x + 5\)[/tex].
To factor the quadratic, look for two numbers that multiply to [tex]\(12 \times 5 = 60\)[/tex] and add up to [tex]\(-19\)[/tex].
The numbers that work are [tex]\(-15\)[/tex] and [tex]\(-4\)[/tex].
3. Rewrite and Factor by Grouping:
Rewrite [tex]\(-19x\)[/tex] as [tex]\(-15x - 4x\)[/tex] and then use factoring by grouping:
[tex]\[
12x^2 - 19x + 5 = 12x^2 - 15x - 4x + 5
\][/tex]
Group the terms:
[tex]\[
(12x^2 - 15x) + (-4x + 5)
\][/tex]
Factor out the GCF from each group:
[tex]\[
3x(4x - 5) - 1(4x - 5)
\][/tex]
Now factor out the common factor [tex]\((4x - 5)\)[/tex]:
[tex]\[
(3x - 1)(4x - 5)
\][/tex]
4. Combine the Factors:
Substitute back into the expression:
[tex]\[
x(12x^2 - 19x + 5) = x(3x - 1)(4x - 5)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(12x^3 - 19x^2 + 5x\)[/tex] is:
[tex]\[
x(3x - 1)(4x - 5)
\][/tex]
