Answer :
Let's factor the polynomial [tex]\(24x^3 + 28x^2 - 24x - 28\)[/tex] completely. We'll go through the steps to find the factors systematically:
1. Group the Terms: Start by grouping the terms in pairs:
[tex]\[ 24x^3 + 28x^2 - 24x - 28 = (24x^3 + 28x^2) + (-24x - 28) \][/tex]
2. Factor by Grouping:
- In the first group, [tex]\(24x^3 + 28x^2\)[/tex], factor out the greatest common factor (GCF), which is [tex]\(4x^2\)[/tex]:
[tex]\[ 24x^3 + 28x^2 = 4x^2(6x + 7) \][/tex]
- In the second group, [tex]\(-24x - 28\)[/tex], factor out the GCF, which is [tex]\(-4\)[/tex]:
[tex]\[ -24x - 28 = -4(6x + 7) \][/tex]
3. Combine the Groups: Now, observe that both groups contain a common factor, [tex]\((6x + 7)\)[/tex]:
[tex]\[ 24x^3 + 28x^2 - 24x - 28 = 4x^2(6x + 7) - 4(6x + 7) \][/tex]
4. Factor Out the Common Binomial:
[tex]\[ 24x^3 + 28x^2 - 24x - 28 = (4x^2 - 4)(6x + 7) \][/tex]
5. Further Factor the Monomial: The expression [tex]\(4x^2 - 4\)[/tex] can also be factored:
- Notice that the GCF of [tex]\(4x^2\)[/tex] and [tex]\(-4\)[/tex] is [tex]\(4\)[/tex]:
[tex]\[ 4x^2 - 4 = 4(x^2 - 1) \][/tex]
6. Recognize a Difference of Squares: The expression [tex]\(x^2 - 1\)[/tex] is a difference of squares:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
7. Combine All Factors:
- Substitute the factored form of [tex]\(x^2 - 1\)[/tex] back into the expression:
[tex]\[ 4(x - 1)(x + 1)(6x + 7) \][/tex]
Thus, the polynomial [tex]\(24x^3 + 28x^2 - 24x - 28\)[/tex] factors completely into:
[tex]\[ 4(x - 1)(x + 1)(6x + 7) \][/tex]
1. Group the Terms: Start by grouping the terms in pairs:
[tex]\[ 24x^3 + 28x^2 - 24x - 28 = (24x^3 + 28x^2) + (-24x - 28) \][/tex]
2. Factor by Grouping:
- In the first group, [tex]\(24x^3 + 28x^2\)[/tex], factor out the greatest common factor (GCF), which is [tex]\(4x^2\)[/tex]:
[tex]\[ 24x^3 + 28x^2 = 4x^2(6x + 7) \][/tex]
- In the second group, [tex]\(-24x - 28\)[/tex], factor out the GCF, which is [tex]\(-4\)[/tex]:
[tex]\[ -24x - 28 = -4(6x + 7) \][/tex]
3. Combine the Groups: Now, observe that both groups contain a common factor, [tex]\((6x + 7)\)[/tex]:
[tex]\[ 24x^3 + 28x^2 - 24x - 28 = 4x^2(6x + 7) - 4(6x + 7) \][/tex]
4. Factor Out the Common Binomial:
[tex]\[ 24x^3 + 28x^2 - 24x - 28 = (4x^2 - 4)(6x + 7) \][/tex]
5. Further Factor the Monomial: The expression [tex]\(4x^2 - 4\)[/tex] can also be factored:
- Notice that the GCF of [tex]\(4x^2\)[/tex] and [tex]\(-4\)[/tex] is [tex]\(4\)[/tex]:
[tex]\[ 4x^2 - 4 = 4(x^2 - 1) \][/tex]
6. Recognize a Difference of Squares: The expression [tex]\(x^2 - 1\)[/tex] is a difference of squares:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
7. Combine All Factors:
- Substitute the factored form of [tex]\(x^2 - 1\)[/tex] back into the expression:
[tex]\[ 4(x - 1)(x + 1)(6x + 7) \][/tex]
Thus, the polynomial [tex]\(24x^3 + 28x^2 - 24x - 28\)[/tex] factors completely into:
[tex]\[ 4(x - 1)(x + 1)(6x + 7) \][/tex]
