Answer :
Sure! Let's go through the process of factoring the polynomial [tex]\(16x^3 + 4x^2 - 100x - 25\)[/tex] completely by grouping:
1. Group the terms:
We start by grouping terms that can be factored together. Here, we can group them as follows:
[tex]\((16x^3 + 4x^2) + (-100x - 25)\)[/tex].
2. Factor each group:
- For the first group [tex]\(16x^3 + 4x^2\)[/tex], the common factor is [tex]\(4x^2\)[/tex]. Factoring this out, we get:
[tex]\(4x^2(4x + 1)\)[/tex].
- For the second group [tex]\(-100x - 25\)[/tex], the common factor is [tex]\(-25\)[/tex]. Factoring this out, we get:
[tex]\(-25(4x + 1)\)[/tex].
3. Combine the factored groups:
Now, notice that both groups contain a common binomial factor [tex]\((4x + 1)\)[/tex]. We can factor this out:
[tex]\((4x + 1)(4x^2 - 25)\)[/tex].
4. Factor further if possible:
The expression [tex]\(4x^2 - 25\)[/tex] is a difference of squares which can be factored using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
- Here, [tex]\(4x^2\)[/tex] is [tex]\((2x)^2\)[/tex] and [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex], so [tex]\(4x^2 - 25\)[/tex] factors into:
[tex]\((2x - 5)(2x + 5)\)[/tex].
5. Final factored form:
Putting it all together, the completely factored form of the polynomial is:
[tex]\((4x + 1)(2x - 5)(2x + 5)\)[/tex].
That’s the complete factorization of the polynomial [tex]\(16x^3 + 4x^2 - 100x - 25\)[/tex].
1. Group the terms:
We start by grouping terms that can be factored together. Here, we can group them as follows:
[tex]\((16x^3 + 4x^2) + (-100x - 25)\)[/tex].
2. Factor each group:
- For the first group [tex]\(16x^3 + 4x^2\)[/tex], the common factor is [tex]\(4x^2\)[/tex]. Factoring this out, we get:
[tex]\(4x^2(4x + 1)\)[/tex].
- For the second group [tex]\(-100x - 25\)[/tex], the common factor is [tex]\(-25\)[/tex]. Factoring this out, we get:
[tex]\(-25(4x + 1)\)[/tex].
3. Combine the factored groups:
Now, notice that both groups contain a common binomial factor [tex]\((4x + 1)\)[/tex]. We can factor this out:
[tex]\((4x + 1)(4x^2 - 25)\)[/tex].
4. Factor further if possible:
The expression [tex]\(4x^2 - 25\)[/tex] is a difference of squares which can be factored using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
- Here, [tex]\(4x^2\)[/tex] is [tex]\((2x)^2\)[/tex] and [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex], so [tex]\(4x^2 - 25\)[/tex] factors into:
[tex]\((2x - 5)(2x + 5)\)[/tex].
5. Final factored form:
Putting it all together, the completely factored form of the polynomial is:
[tex]\((4x + 1)(2x - 5)(2x + 5)\)[/tex].
That’s the complete factorization of the polynomial [tex]\(16x^3 + 4x^2 - 100x - 25\)[/tex].
