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Answer :
To factor the polynomial [tex]\(28x^3 - 48x^2 + 28x - 48\)[/tex] completely, follow these steps:
1. Look for a common factor among all the terms.
- The numbers 28, 48, 28, and 48 all have a common factor of 4. So, first, factor out 4:
[tex]\[
4(7x^3 - 12x^2 + 7x - 12)
\][/tex]
2. Factor the remaining polynomial [tex]\(7x^3 - 12x^2 + 7x - 12\)[/tex].
- To factor this polynomial, we can use grouping or try to find patterns like common terms in specific pairs.
3. Check for further factorization within the cubic expression.
- Observe [tex]\(7x^3 - 12x^2 + 7x - 12\)[/tex]. We will try to find how we can factor this further. Notice that there is no obvious common factor within half of the terms, so let's try a different approach by treating the expression as a pair:
Grouping:
[tex]\[
(7x^3 - 12x^2) + (7x - 12)
\][/tex]
Factor out common factors from each group:
- From [tex]\(7x^3 - 12x^2\)[/tex], factor out [tex]\(x^2\)[/tex]: [tex]\((7x - 12)x^2\)[/tex]
- From [tex]\(7x - 12\)[/tex], there is no further factorization within integer limits: [tex]\(7x - 12\)[/tex]
4. Form the final factored expression:
When looking at the expression again, another approach reveals that the factorization results in quadratic expression as:
[tex]\[
4(7x - 12)(x^2 + 1)
\][/tex]
5. Result:
So, the polynomial [tex]\(28x^3 - 48x^2 + 28x - 48\)[/tex] completely factors to:
[tex]\[
4(7x - 12)(x^2 + 1)
\][/tex]
This is the completely factored form. Each step ensures that the solution is accurate and neatly laid out for easy verification.
1. Look for a common factor among all the terms.
- The numbers 28, 48, 28, and 48 all have a common factor of 4. So, first, factor out 4:
[tex]\[
4(7x^3 - 12x^2 + 7x - 12)
\][/tex]
2. Factor the remaining polynomial [tex]\(7x^3 - 12x^2 + 7x - 12\)[/tex].
- To factor this polynomial, we can use grouping or try to find patterns like common terms in specific pairs.
3. Check for further factorization within the cubic expression.
- Observe [tex]\(7x^3 - 12x^2 + 7x - 12\)[/tex]. We will try to find how we can factor this further. Notice that there is no obvious common factor within half of the terms, so let's try a different approach by treating the expression as a pair:
Grouping:
[tex]\[
(7x^3 - 12x^2) + (7x - 12)
\][/tex]
Factor out common factors from each group:
- From [tex]\(7x^3 - 12x^2\)[/tex], factor out [tex]\(x^2\)[/tex]: [tex]\((7x - 12)x^2\)[/tex]
- From [tex]\(7x - 12\)[/tex], there is no further factorization within integer limits: [tex]\(7x - 12\)[/tex]
4. Form the final factored expression:
When looking at the expression again, another approach reveals that the factorization results in quadratic expression as:
[tex]\[
4(7x - 12)(x^2 + 1)
\][/tex]
5. Result:
So, the polynomial [tex]\(28x^3 - 48x^2 + 28x - 48\)[/tex] completely factors to:
[tex]\[
4(7x - 12)(x^2 + 1)
\][/tex]
This is the completely factored form. Each step ensures that the solution is accurate and neatly laid out for easy verification.
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