Answer :
Sure, let's factor the expression completely by first factoring out the greatest common factor (GCF).
The expression we have is:
[tex]\[ 12x^3 + 8x^2 - 32x \][/tex]
Step 1: Identify the Greatest Common Factor (GCF)
Look for the GCF of the coefficients and the variable part in each term.
- The coefficients are 12, 8, and -32. The greatest common factor of these numbers is 4.
- For the variable [tex]\(x\)[/tex], each term has at least one [tex]\(x\)[/tex]. Therefore, the GCF for the variables is [tex]\(x\)[/tex].
So, the GCF of the entire expression is [tex]\(4x\)[/tex].
Step 2: Factor out the GCF from each term
Divide each term in the expression by the GCF ([tex]\(4x\)[/tex]):
[tex]\[ 12x^3 \div 4x = 3x^2 \][/tex]
[tex]\[ 8x^2 \div 4x = 2x \][/tex]
[tex]\[ -32x \div 4x = -8 \][/tex]
Now, factor the expression using the GCF:
[tex]\[ 12x^3 + 8x^2 - 32x = 4x(3x^2 + 2x - 8) \][/tex]
Step 3: Factor the Quadratic Trinomial
Now, we need to factor the quadratic trinomial [tex]\(3x^2 + 2x - 8\)[/tex].
To factor this, we'll look for two numbers that multiply to [tex]\(3 \times -8 = -24\)[/tex] and add to [tex]\(2\)[/tex].
The numbers that satisfy this are [tex]\(6\)[/tex] and [tex]\(-4\)[/tex].
Rewrite the middle term using these numbers:
[tex]\[ 3x^2 + 6x - 4x - 8 \][/tex]
Group the terms:
[tex]\[ (3x^2 + 6x) + (-4x - 8) \][/tex]
Factor by grouping:
[tex]\[ 3x(x + 2) - 4(x + 2) \][/tex]
Notice that [tex]\((x + 2)\)[/tex] is a common factor:
[tex]\[ (3x - 4)(x + 2) \][/tex]
Step 4: Combine all factors
Finally, substitute back into the expression with the GCF factored out:
[tex]\[ 4x(3x^2 + 2x - 8) \][/tex]
This simplifies to:
[tex]\[ 4x(x + 2)(3x - 4) \][/tex]
So, the completely factored form of [tex]\(12x^3 + 8x^2 - 32x\)[/tex] is:
[tex]\[ 4x(x + 2)(3x - 4) \][/tex]
The expression we have is:
[tex]\[ 12x^3 + 8x^2 - 32x \][/tex]
Step 1: Identify the Greatest Common Factor (GCF)
Look for the GCF of the coefficients and the variable part in each term.
- The coefficients are 12, 8, and -32. The greatest common factor of these numbers is 4.
- For the variable [tex]\(x\)[/tex], each term has at least one [tex]\(x\)[/tex]. Therefore, the GCF for the variables is [tex]\(x\)[/tex].
So, the GCF of the entire expression is [tex]\(4x\)[/tex].
Step 2: Factor out the GCF from each term
Divide each term in the expression by the GCF ([tex]\(4x\)[/tex]):
[tex]\[ 12x^3 \div 4x = 3x^2 \][/tex]
[tex]\[ 8x^2 \div 4x = 2x \][/tex]
[tex]\[ -32x \div 4x = -8 \][/tex]
Now, factor the expression using the GCF:
[tex]\[ 12x^3 + 8x^2 - 32x = 4x(3x^2 + 2x - 8) \][/tex]
Step 3: Factor the Quadratic Trinomial
Now, we need to factor the quadratic trinomial [tex]\(3x^2 + 2x - 8\)[/tex].
To factor this, we'll look for two numbers that multiply to [tex]\(3 \times -8 = -24\)[/tex] and add to [tex]\(2\)[/tex].
The numbers that satisfy this are [tex]\(6\)[/tex] and [tex]\(-4\)[/tex].
Rewrite the middle term using these numbers:
[tex]\[ 3x^2 + 6x - 4x - 8 \][/tex]
Group the terms:
[tex]\[ (3x^2 + 6x) + (-4x - 8) \][/tex]
Factor by grouping:
[tex]\[ 3x(x + 2) - 4(x + 2) \][/tex]
Notice that [tex]\((x + 2)\)[/tex] is a common factor:
[tex]\[ (3x - 4)(x + 2) \][/tex]
Step 4: Combine all factors
Finally, substitute back into the expression with the GCF factored out:
[tex]\[ 4x(3x^2 + 2x - 8) \][/tex]
This simplifies to:
[tex]\[ 4x(x + 2)(3x - 4) \][/tex]
So, the completely factored form of [tex]\(12x^3 + 8x^2 - 32x\)[/tex] is:
[tex]\[ 4x(x + 2)(3x - 4) \][/tex]
