Answer :
To solve this problem, we need to factor the expression [tex]\(8x^5 - 4x^4 - 72x^3 + 60x^2\)[/tex].
Let's look at the expression step-by-step:
1. Identify the Greatest Common Factor (GCF):
First, we identify the greatest common factor of all the terms. The coefficients are 8, -4, -72, and 60. The greatest common factor of these numbers is 4.
For the variable part, each term has a factor of [tex]\(x^2\)[/tex]. So, the GCF of the entire polynomial is [tex]\(4x^2\)[/tex].
2. Factor out the GCF:
When we factor [tex]\(4x^2\)[/tex] out of each term, we get:
[tex]\[
4x^2(2x^3 - x^2 - 18x + 15)
\][/tex]
3. Verify with the given options:
Now, we need to select the appropriate factors from the given options to complete the expression. From the steps above, we determined that the GCF was [tex]\(4x^2\)[/tex], and the remaining polynomial was [tex]\(2x^3 - x^2 - 18x + 15\)[/tex].
4. Choose the appropriate terms:
Looking at our factored form [tex]\(4x^2(2x^3 - x^2 - 18x + 15)\)[/tex], we can match each part with the supplied options. To represent the expression, you would need:
- 4 (The GCF for coefficients)
- [tex]\(x^2\)[/tex] (The common factor of the smallest power of x)
- 15 (A coefficient inside the polynomial factor)
- [tex]\(x^3\)[/tex] (One of the terms with x in the polynomial)
So, the parts needed to complete the expression based on the options provided are:
1. 4
2. [tex]\(x^2\)[/tex]
3. 15
4. [tex]\(x^3\)[/tex]
Therefore, the correct complete expression using these options is:
[tex]\[ 4, x^2, 15, x^3 \][/tex]
This matches the structure we are aiming to express in the factorized polynomial.
Let's look at the expression step-by-step:
1. Identify the Greatest Common Factor (GCF):
First, we identify the greatest common factor of all the terms. The coefficients are 8, -4, -72, and 60. The greatest common factor of these numbers is 4.
For the variable part, each term has a factor of [tex]\(x^2\)[/tex]. So, the GCF of the entire polynomial is [tex]\(4x^2\)[/tex].
2. Factor out the GCF:
When we factor [tex]\(4x^2\)[/tex] out of each term, we get:
[tex]\[
4x^2(2x^3 - x^2 - 18x + 15)
\][/tex]
3. Verify with the given options:
Now, we need to select the appropriate factors from the given options to complete the expression. From the steps above, we determined that the GCF was [tex]\(4x^2\)[/tex], and the remaining polynomial was [tex]\(2x^3 - x^2 - 18x + 15\)[/tex].
4. Choose the appropriate terms:
Looking at our factored form [tex]\(4x^2(2x^3 - x^2 - 18x + 15)\)[/tex], we can match each part with the supplied options. To represent the expression, you would need:
- 4 (The GCF for coefficients)
- [tex]\(x^2\)[/tex] (The common factor of the smallest power of x)
- 15 (A coefficient inside the polynomial factor)
- [tex]\(x^3\)[/tex] (One of the terms with x in the polynomial)
So, the parts needed to complete the expression based on the options provided are:
1. 4
2. [tex]\(x^2\)[/tex]
3. 15
4. [tex]\(x^3\)[/tex]
Therefore, the correct complete expression using these options is:
[tex]\[ 4, x^2, 15, x^3 \][/tex]
This matches the structure we are aiming to express in the factorized polynomial.
