Answer :
Sure! Let's factor out the greatest common factor (GCF) in the expression [tex]\(3x^4 - 21x^3 + 33x^2\)[/tex].
### Step-by-Step Solution:
1. Identify the GCF of the Coefficients:
- The coefficients of the terms are 3, 21, and 33.
- Find the GCF of these numbers:
- 3 can be divided by 3.
- 21 can be divided by 3.
- 33 can be divided by 3.
- So, the GCF of 3, 21, and 33 is 3.
2. Identify the GCF of the Variable Part:
- The expression has powers of [tex]\(x\)[/tex]: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power is [tex]\(x^2\)[/tex].
- Therefore, the GCF for the variable part is [tex]\(x^2\)[/tex].
3. Combine the GCFs:
- The overall GCF of the expression is [tex]\(3x^2\)[/tex].
4. Factor Out the GCF:
- Divide each term in the expression by the GCF [tex]\(3x^2\)[/tex]:
- [tex]\( \frac{3x^4}{3x^2} = x^2 \)[/tex]
- [tex]\( \frac{-21x^3}{3x^2} = -7x \)[/tex]
- [tex]\( \frac{33x^2}{3x^2} = 11 \)[/tex]
- After factoring out the GCF, the expression inside is [tex]\(x^2 - 7x + 11\)[/tex].
5. Write the Final Expression:
- The factored expression is: [tex]\(3x^2(x^2 - 7x + 11)\)[/tex].
So, the factorization of the expression [tex]\(3x^4 - 21x^3 + 33x^2\)[/tex] is [tex]\(3x^2(x^2 - 7x + 11)\)[/tex].
### Step-by-Step Solution:
1. Identify the GCF of the Coefficients:
- The coefficients of the terms are 3, 21, and 33.
- Find the GCF of these numbers:
- 3 can be divided by 3.
- 21 can be divided by 3.
- 33 can be divided by 3.
- So, the GCF of 3, 21, and 33 is 3.
2. Identify the GCF of the Variable Part:
- The expression has powers of [tex]\(x\)[/tex]: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex].
- The smallest power is [tex]\(x^2\)[/tex].
- Therefore, the GCF for the variable part is [tex]\(x^2\)[/tex].
3. Combine the GCFs:
- The overall GCF of the expression is [tex]\(3x^2\)[/tex].
4. Factor Out the GCF:
- Divide each term in the expression by the GCF [tex]\(3x^2\)[/tex]:
- [tex]\( \frac{3x^4}{3x^2} = x^2 \)[/tex]
- [tex]\( \frac{-21x^3}{3x^2} = -7x \)[/tex]
- [tex]\( \frac{33x^2}{3x^2} = 11 \)[/tex]
- After factoring out the GCF, the expression inside is [tex]\(x^2 - 7x + 11\)[/tex].
5. Write the Final Expression:
- The factored expression is: [tex]\(3x^2(x^2 - 7x + 11)\)[/tex].
So, the factorization of the expression [tex]\(3x^4 - 21x^3 + 33x^2\)[/tex] is [tex]\(3x^2(x^2 - 7x + 11)\)[/tex].
