Answer :
Sure! Let's go through how you can factor each of these expressions step-by-step:
a. [tex]\(3x^3 + 18x^2 + 9x - 30\)[/tex]
1. Factor out the greatest common factor (GCF), which is 3:
[tex]\[
3(x^3 + 6x^2 + 3x - 10)
\][/tex]
2. Find the roots of the cubic polynomial [tex]\(x^3 + 6x^2 + 3x - 10\)[/tex] using trial, error, or synthetic division, finding [tex]\(x = 1\)[/tex] as a root:
[tex]\[
3((x - 1)(x^2 + 7x + 10))
\][/tex]
3. Factor [tex]\(x^2 + 7x + 10\)[/tex] as [tex]\((x + 2)(x + 5)\)[/tex]:
[tex]\[
3(x - 1)(x + 2)(x + 5)
\][/tex]
b. [tex]\(x^3 + 6x^2 + 4x + 24\)[/tex]
1. Notice [tex]\(x^3\)[/tex] and [tex]\(6x^2\)[/tex] can be grouped separately from [tex]\(4x + 24\)[/tex]:
[tex]\[
x^2(x + 6) + 4(x + 6)
\][/tex]
2. Factor by grouping:
[tex]\[
(x^2 + 4)(x + 6)
\][/tex]
c. [tex]\(8x^3 + 27\)[/tex]
1. Recognize this as a sum of cubes, [tex]\( (2x)^3 + 3^3 \)[/tex]:
2. Use the sum of cubes formula, [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]:
[tex]\[
(2x + 3)(4x^2 - 6x + 9)
\][/tex]
d. [tex]\(x^6 - y^6\)[/tex]
1. Recognize this as a difference of squares:
[tex]\[
(x^3)^2 - (y^3)^2
\][/tex]
2. Use the difference of squares formula, [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
(x^3 - y^3)(x^3 + y^3)
\][/tex]
3. Factor each resulting cubic using the sum or difference of cubes formulas:
[tex]\[
(x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2)
\][/tex]
e. [tex]\(6x^4 - 28x^3 - 25x^2 + 70x + 25\)[/tex]
1. Find one real root, using trial, error, or synthetic division, such as [tex]\(x = 5\)[/tex]:
[tex]\[
(x - 5)(\text{(resulting cubic polynomial)})
\][/tex]
2. Factor the quadratic polynomial:
[tex]\[
(3x + 1)(2x^2 - 5)
\][/tex]
3. Combine into:
[tex]\[
(x - 5)(3x + 1)(2x^2 - 5)
\][/tex]
f. [tex]\(4x^3 + 12x^2 - 25x - 75\)[/tex]
1. Factor out the greatest common factor (GCF), which is 1, and proceed with factorization:
2. Find a root using trial, error, or synthetic division, such as [tex]\(x = -3\)[/tex]:
[tex]\[
(x + 3)(\text{(resulting quadratic polynomial)})
\][/tex]
3. Factor the quadratic polynomial:
[tex]\[
(2x - 5)(2x + 5)
\][/tex]
4. Combine into:
[tex]\[
(x + 3)(2x - 5)(2x + 5)
\][/tex]
I hope this helps you understand how to factor these expressions!
a. [tex]\(3x^3 + 18x^2 + 9x - 30\)[/tex]
1. Factor out the greatest common factor (GCF), which is 3:
[tex]\[
3(x^3 + 6x^2 + 3x - 10)
\][/tex]
2. Find the roots of the cubic polynomial [tex]\(x^3 + 6x^2 + 3x - 10\)[/tex] using trial, error, or synthetic division, finding [tex]\(x = 1\)[/tex] as a root:
[tex]\[
3((x - 1)(x^2 + 7x + 10))
\][/tex]
3. Factor [tex]\(x^2 + 7x + 10\)[/tex] as [tex]\((x + 2)(x + 5)\)[/tex]:
[tex]\[
3(x - 1)(x + 2)(x + 5)
\][/tex]
b. [tex]\(x^3 + 6x^2 + 4x + 24\)[/tex]
1. Notice [tex]\(x^3\)[/tex] and [tex]\(6x^2\)[/tex] can be grouped separately from [tex]\(4x + 24\)[/tex]:
[tex]\[
x^2(x + 6) + 4(x + 6)
\][/tex]
2. Factor by grouping:
[tex]\[
(x^2 + 4)(x + 6)
\][/tex]
c. [tex]\(8x^3 + 27\)[/tex]
1. Recognize this as a sum of cubes, [tex]\( (2x)^3 + 3^3 \)[/tex]:
2. Use the sum of cubes formula, [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]:
[tex]\[
(2x + 3)(4x^2 - 6x + 9)
\][/tex]
d. [tex]\(x^6 - y^6\)[/tex]
1. Recognize this as a difference of squares:
[tex]\[
(x^3)^2 - (y^3)^2
\][/tex]
2. Use the difference of squares formula, [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]:
[tex]\[
(x^3 - y^3)(x^3 + y^3)
\][/tex]
3. Factor each resulting cubic using the sum or difference of cubes formulas:
[tex]\[
(x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2)
\][/tex]
e. [tex]\(6x^4 - 28x^3 - 25x^2 + 70x + 25\)[/tex]
1. Find one real root, using trial, error, or synthetic division, such as [tex]\(x = 5\)[/tex]:
[tex]\[
(x - 5)(\text{(resulting cubic polynomial)})
\][/tex]
2. Factor the quadratic polynomial:
[tex]\[
(3x + 1)(2x^2 - 5)
\][/tex]
3. Combine into:
[tex]\[
(x - 5)(3x + 1)(2x^2 - 5)
\][/tex]
f. [tex]\(4x^3 + 12x^2 - 25x - 75\)[/tex]
1. Factor out the greatest common factor (GCF), which is 1, and proceed with factorization:
2. Find a root using trial, error, or synthetic division, such as [tex]\(x = -3\)[/tex]:
[tex]\[
(x + 3)(\text{(resulting quadratic polynomial)})
\][/tex]
3. Factor the quadratic polynomial:
[tex]\[
(2x - 5)(2x + 5)
\][/tex]
4. Combine into:
[tex]\[
(x + 3)(2x - 5)(2x + 5)
\][/tex]
I hope this helps you understand how to factor these expressions!
