Answer :
Certainly! Let's factor each polynomial completely, step by step:
1. Factor [tex]\( 12x^3 + 96x^2 + 192x \)[/tex]:
- Step 1: Identify the greatest common factor (GCF) of all terms. The GCF here is [tex]\( 12x \)[/tex].
- Step 2: Factor out the GCF from the polynomial:
[tex]\[
12x(x^2 + 8x + 16)
\][/tex]
- Step 3: Notice that the quadratic [tex]\( x^2 + 8x + 16 \)[/tex] is a perfect square trinomial:
[tex]\[
x^2 + 8x + 16 = (x + 4)^2
\][/tex]
- Final Factored Form:
[tex]\[
12x(x + 4)^2
\][/tex]
2. Factor [tex]\( 108x^6 + 32x^3 \)[/tex]:
- Step 1: Identify the GCF, which is [tex]\( 4x^3 \)[/tex].
- Step 2: Factor out the GCF:
[tex]\[
4x^3(27x^3 + 8)
\][/tex]
- Step 3: Recognize [tex]\( 27x^3 + 8 \)[/tex] as a sum of cubes:
[tex]\[
27x^3 + 8 = (3x)^3 + 2^3
\][/tex]
- Step 4: Use the sum of cubes formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
For [tex]\( a = 3x \)[/tex] and [tex]\( b = 2 \)[/tex], it becomes:
[tex]\[
(3x + 2)((3x)^2 - (3x)(2) + 2^2) = (3x + 2)(9x^2 - 6x + 4)
\][/tex]
- Final Factored Form:
[tex]\[
4x^3(3x + 2)(9x^2 - 6x + 4)
\][/tex]
3. Factor [tex]\( x^3 + x^2 - 16x - 16 \)[/tex]:
- Step 1: Use factoring by grouping. Pair the terms:
[tex]\[
(x^3 + x^2) + (-16x - 16)
\][/tex]
- Step 2: Factor each pair:
[tex]\[
x^2(x + 1) - 16(x + 1)
\][/tex]
- Step 3: Notice the common factor [tex]\((x + 1)\)[/tex]:
[tex]\[
(x + 1)(x^2 - 16)
\][/tex]
- Step 4: Recognize [tex]\( x^2 - 16 \)[/tex] as a difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
- Final Factored Form:
[tex]\[
(x + 1)(x - 4)(x + 4)
\][/tex]
4. Factor [tex]\( 7x^{12} + 49x^9 + 70x^6 \)[/tex]:
- Step 1: Find the GCF, which is [tex]\( 7x^6 \)[/tex].
- Step 2: Factor out the GCF:
[tex]\[
7x^6(x^6 + 7x^3 + 10)
\][/tex]
- Step 3: Notice further factoring is necessary for [tex]\( x^6 + 7x^3 + 10 \)[/tex], but considering stem structure or complexity, it results in:
[tex]\[
(x^3 + 2)(x^3 + 5)
\][/tex]
- Final Factored Form:
[tex]\[
7x^6(x^3 + 2)(x^3 + 5)
\][/tex]
These steps will help you understand how the polynomials were factored completely.
1. Factor [tex]\( 12x^3 + 96x^2 + 192x \)[/tex]:
- Step 1: Identify the greatest common factor (GCF) of all terms. The GCF here is [tex]\( 12x \)[/tex].
- Step 2: Factor out the GCF from the polynomial:
[tex]\[
12x(x^2 + 8x + 16)
\][/tex]
- Step 3: Notice that the quadratic [tex]\( x^2 + 8x + 16 \)[/tex] is a perfect square trinomial:
[tex]\[
x^2 + 8x + 16 = (x + 4)^2
\][/tex]
- Final Factored Form:
[tex]\[
12x(x + 4)^2
\][/tex]
2. Factor [tex]\( 108x^6 + 32x^3 \)[/tex]:
- Step 1: Identify the GCF, which is [tex]\( 4x^3 \)[/tex].
- Step 2: Factor out the GCF:
[tex]\[
4x^3(27x^3 + 8)
\][/tex]
- Step 3: Recognize [tex]\( 27x^3 + 8 \)[/tex] as a sum of cubes:
[tex]\[
27x^3 + 8 = (3x)^3 + 2^3
\][/tex]
- Step 4: Use the sum of cubes formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
For [tex]\( a = 3x \)[/tex] and [tex]\( b = 2 \)[/tex], it becomes:
[tex]\[
(3x + 2)((3x)^2 - (3x)(2) + 2^2) = (3x + 2)(9x^2 - 6x + 4)
\][/tex]
- Final Factored Form:
[tex]\[
4x^3(3x + 2)(9x^2 - 6x + 4)
\][/tex]
3. Factor [tex]\( x^3 + x^2 - 16x - 16 \)[/tex]:
- Step 1: Use factoring by grouping. Pair the terms:
[tex]\[
(x^3 + x^2) + (-16x - 16)
\][/tex]
- Step 2: Factor each pair:
[tex]\[
x^2(x + 1) - 16(x + 1)
\][/tex]
- Step 3: Notice the common factor [tex]\((x + 1)\)[/tex]:
[tex]\[
(x + 1)(x^2 - 16)
\][/tex]
- Step 4: Recognize [tex]\( x^2 - 16 \)[/tex] as a difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
- Final Factored Form:
[tex]\[
(x + 1)(x - 4)(x + 4)
\][/tex]
4. Factor [tex]\( 7x^{12} + 49x^9 + 70x^6 \)[/tex]:
- Step 1: Find the GCF, which is [tex]\( 7x^6 \)[/tex].
- Step 2: Factor out the GCF:
[tex]\[
7x^6(x^6 + 7x^3 + 10)
\][/tex]
- Step 3: Notice further factoring is necessary for [tex]\( x^6 + 7x^3 + 10 \)[/tex], but considering stem structure or complexity, it results in:
[tex]\[
(x^3 + 2)(x^3 + 5)
\][/tex]
- Final Factored Form:
[tex]\[
7x^6(x^3 + 2)(x^3 + 5)
\][/tex]
These steps will help you understand how the polynomials were factored completely.
