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------------------------------------------------ (IV) Factor the following equations completely.

1) [tex]28x^3 + 49x^2 - 16x - 28[/tex]

2) [tex]16g^4 - 625[/tex]

3) [tex]64x^3 + 27[/tex]

4) [tex]2x^4 - 12x^3 + 18x^2[/tex]

Answer :

Sure! Let's factor each of these expressions step by step.

1) Factor [tex]\(28x^3 + 49x^2 - 16x - 28\)[/tex]:

First, we can group and factor by grouping:
[tex]\[
(28x^3 + 49x^2) + (-16x - 28)
\][/tex]

Factor out the common factor from each group:
[tex]\[
7x^2(4x + 7) - 4(4x + 7)
\][/tex]

Now, notice that [tex]\(4x + 7\)[/tex] is a common factor:
[tex]\[
(7x^2 - 4)(4x + 7)
\][/tex]

This is the completely factored form.

2) Factor [tex]\(16g^4 - 625\)[/tex]:

This is a difference of squares:
[tex]\[
(4g^2)^2 - 25^2
\][/tex]

Apply the difference of squares formula: [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
(4g^2 - 25)(4g^2 + 25)
\][/tex]

The first factor, [tex]\(4g^2 - 25\)[/tex], is also a difference of squares:
[tex]\[
(2g - 5)(2g + 5)
\][/tex]

Thus, the complete factorization is:
[tex]\[
(2g - 5)(2g + 5)(4g^2 + 25)
\][/tex]

3) Factor [tex]\(64x^3 + 27\)[/tex]:

This is a sum of cubes:
[tex]\[
(4x)^3 + 3^3
\][/tex]

Apply the sum of cubes formula: [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]:
[tex]\[
(4x + 3)((4x)^2 - (4x)(3) + 3^2)
\][/tex]

Simplify:
[tex]\[
(4x + 3)(16x^2 - 12x + 9)
\][/tex]

4) Factor [tex]\(2x^4 - 12x^3 + 18x^2\)[/tex]:

First, factor out the greatest common factor:
[tex]\[
2x^2(x^2 - 6x + 9)
\][/tex]

Next, notice that [tex]\(x^2 - 6x + 9\)[/tex] is a perfect square trinomial:
[tex]\[
(x - 3)^2
\][/tex]

Therefore, the complete factorization is:
[tex]\[
2x^2(x - 3)^2
\][/tex]

That completes the factorization of all given expressions! Let me know if you need further clarification or help.