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Answer :
To factor out the greatest common factor (GCF) in the expression [tex]\(7x^4 - 14x^3 + 21x^2\)[/tex], here's how you can do it step-by-step:
1. Identify the GCF for the coefficients:
- Look at the numbers: 7, 14, and 21.
- The GCF of these numbers is 7 because 7 is the largest number that can divide each of them without leaving a remainder.
2. Identify the smallest power of [tex]\(x\)[/tex]:
- Consider the powers of [tex]\(x\)[/tex] in each term: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
3. Combine the GCF:
- The GCF for the whole expression factoring in both the numerical coefficient and the variable is [tex]\(7x^2\)[/tex].
4. Factor out the GCF from each term:
- Divide each term in the expression by [tex]\(7x^2\)[/tex]:
- For [tex]\(7x^4\)[/tex], divide by [tex]\(7x^2\)[/tex]: [tex]\((7x^4) \div (7x^2) = x^2\)[/tex].
- For [tex]\(-14x^3\)[/tex], divide by [tex]\(7x^2\)[/tex]: [tex]\((-14x^3) \div (7x^2) = -2x\)[/tex].
- For [tex]\(21x^2\)[/tex], divide by [tex]\(7x^2\)[/tex]: [tex]\((21x^2) \div (7x^2) = 3\)[/tex].
5. Write the factored expression:
- Combine everything into the factored form:
[tex]\[
7x^2(x^2 - 2x + 3)
\][/tex]
So the factored expression is [tex]\(7x^2(x^2 - 2x + 3)\)[/tex]. This means that the greatest common factor has been successfully extracted from the original expression.
1. Identify the GCF for the coefficients:
- Look at the numbers: 7, 14, and 21.
- The GCF of these numbers is 7 because 7 is the largest number that can divide each of them without leaving a remainder.
2. Identify the smallest power of [tex]\(x\)[/tex]:
- Consider the powers of [tex]\(x\)[/tex] in each term: [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], [tex]\(x^2\)[/tex].
- The smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
3. Combine the GCF:
- The GCF for the whole expression factoring in both the numerical coefficient and the variable is [tex]\(7x^2\)[/tex].
4. Factor out the GCF from each term:
- Divide each term in the expression by [tex]\(7x^2\)[/tex]:
- For [tex]\(7x^4\)[/tex], divide by [tex]\(7x^2\)[/tex]: [tex]\((7x^4) \div (7x^2) = x^2\)[/tex].
- For [tex]\(-14x^3\)[/tex], divide by [tex]\(7x^2\)[/tex]: [tex]\((-14x^3) \div (7x^2) = -2x\)[/tex].
- For [tex]\(21x^2\)[/tex], divide by [tex]\(7x^2\)[/tex]: [tex]\((21x^2) \div (7x^2) = 3\)[/tex].
5. Write the factored expression:
- Combine everything into the factored form:
[tex]\[
7x^2(x^2 - 2x + 3)
\][/tex]
So the factored expression is [tex]\(7x^2(x^2 - 2x + 3)\)[/tex]. This means that the greatest common factor has been successfully extracted from the original expression.
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