Answer :
Sure! Let's go through each of the selected problems step by step:
### 3. Factor the difference of squares: [tex]\(121p^2 - 169\)[/tex]
1. Identify the difference of squares: We have [tex]\(121p^2\)[/tex] and [tex]\(169\)[/tex], which are both perfect squares.
2. Write each term as a square: [tex]\(121p^2 = (11p)^2\)[/tex] and [tex]\(169 = 13^2\)[/tex].
3. Apply the formula for the difference of squares:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
In this case, [tex]\(a = 11p\)[/tex] and [tex]\(b = 13\)[/tex].
4. Factor the expression:
[tex]\[
121p^2 - 169 = (11p - 13)(11p + 13)
\][/tex]
### 11. Factor the perfect square: [tex]\(225y^2 + 120y + 16\)[/tex]
1. Check if it’s a perfect square trinomial:
2. The expression can be rewritten as:
[tex]\[
(15y)^2 + 2(15y)(4) + 4^2
\][/tex]
3. This fits the pattern [tex]\(a^2 + 2ab + b^2 = (a + b)^2\)[/tex].
4. Here, [tex]\(a = 15y\)[/tex] and [tex]\(b = 4\)[/tex].
5. Factor the expression:
[tex]\[
225y^2 + 120y + 16 = (15y + 4)^2
\][/tex]
### 19. Factor the difference of cubes: [tex]\(64x^3 - 125\)[/tex]
1. Recognize the difference of cubes:
We have [tex]\(64x^3 = (4x)^3\)[/tex] and [tex]\(125 = 5^3\)[/tex].
2. Use the formula for the difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
In this case, [tex]\(a = 4x\)[/tex] and [tex]\(b = 5\)[/tex].
3. Factor the expression:
[tex]\[
64x^3 - 125 = (4x - 5)((4x)^2 + (4x)(5) + 5^2)
\][/tex]
4. Simplify the second factor:
[tex]\[
(4x)^2 + 20x + 25 = 16x^2 + 20x + 25
\][/tex]
5. Final factorization:
[tex]\[
(4x - 5)(16x^2 + 20x + 25)
\][/tex]
### 26. Factor the quadratic polynomial: [tex]\(20w^2 - 47w + 24\)[/tex]
1. Find two numbers that multiply to [tex]\(20 \times 24 = 480\)[/tex] and add to [tex]\(-47\)[/tex]:
The numbers are [tex]\(-40\)[/tex] and [tex]\(-7\)[/tex].
2. Split the middle term using these numbers:
[tex]\[
20w^2 - 40w - 7w + 24
\][/tex]
3. Factor by grouping:
[tex]\[
20w^2 - 40w - 7w + 24 = (20w^2 - 40w) + (-7w + 24)
\][/tex]
[tex]\[
= 20w(w - 2) - 7(w - 2)
\][/tex]
4. Factor out the common term (w - 2):
[tex]\[
= (20w - 7)(w - 2)
\][/tex]
5. Final factorization:
[tex]\[
= (4w - 3)(5w - 8)
\][/tex]
### 32. Factor the quadratic polynomial: [tex]\(90v^2 - 181v + 90\)[/tex]
1. Find two numbers that multiply to [tex]\(90 \times 90 = 8100\)[/tex] and add to [tex]\(-181\)[/tex]:
The numbers are [tex]\(-90\)[/tex] and [tex]\(-91\)[/tex].
2. Split the middle term using these numbers:
[tex]\[
90v^2 - 90v - 91v + 90
\][/tex]
3. Factor by grouping:
[tex]\[
90v^2 - 90v - 91v + 90 = (90v^2 - 90v) + (-91v + 90)
\][/tex]
[tex]\[
= 90v(v - 1) - 91(v - 1)
\][/tex]
4. Factor out the common term (v - 1):
[tex]\[
= (90v - 91)(v - 1)
\][/tex]
5. Final factorization:
[tex]\[
= (9v - 10)(10v - 9)
\][/tex]
### 36. Completely factor the polynomial: [tex]\(x^3 + x^2 - 20x\)[/tex]
1. Factor out the greatest common factor, [tex]\(x\)[/tex]:
[tex]\[
x(x^2 + x - 20)
\][/tex]
2. Factor the quadratic [tex]\(x^2 + x - 20\)[/tex]:
- Find two numbers that multiply to [tex]\(-20\)[/tex] and add to [tex]\(1\)[/tex]: The numbers are [tex]\(5\)[/tex] and [tex]\(-4\)[/tex].
3. Factor the quadratic:
[tex]\[
x^2 + x - 20 = (x + 5)(x - 4)
\][/tex]
4. Final factorization:
[tex]\[
x(x + 5)(x - 4)
\][/tex]
### 40. Completely factor the polynomial: [tex]\(2x^3 - x^2 - 8x + 4\)[/tex]
1. Look for common factor in pairs:
[tex]\[
(2x^3 - x^2) + (-8x + 4)
\][/tex]
2. Factor each pair:
[tex]\[
x^2(2x - 1) - 4(2x - 1)
\][/tex]
3. Combine the factored terms:
[tex]\[
= (x^2 - 4)(2x - 1)
\][/tex]
4. Factor [tex]\(x^2 - 4\)[/tex] further using difference of squares:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
5. Final factorization:
[tex]\[
(x - 2)(x + 2)(2x - 1)
\][/tex]
These steps cover the detailed factorization process for each selected problem. If you have any more questions or need further clarification, feel free to ask!
### 3. Factor the difference of squares: [tex]\(121p^2 - 169\)[/tex]
1. Identify the difference of squares: We have [tex]\(121p^2\)[/tex] and [tex]\(169\)[/tex], which are both perfect squares.
2. Write each term as a square: [tex]\(121p^2 = (11p)^2\)[/tex] and [tex]\(169 = 13^2\)[/tex].
3. Apply the formula for the difference of squares:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
In this case, [tex]\(a = 11p\)[/tex] and [tex]\(b = 13\)[/tex].
4. Factor the expression:
[tex]\[
121p^2 - 169 = (11p - 13)(11p + 13)
\][/tex]
### 11. Factor the perfect square: [tex]\(225y^2 + 120y + 16\)[/tex]
1. Check if it’s a perfect square trinomial:
2. The expression can be rewritten as:
[tex]\[
(15y)^2 + 2(15y)(4) + 4^2
\][/tex]
3. This fits the pattern [tex]\(a^2 + 2ab + b^2 = (a + b)^2\)[/tex].
4. Here, [tex]\(a = 15y\)[/tex] and [tex]\(b = 4\)[/tex].
5. Factor the expression:
[tex]\[
225y^2 + 120y + 16 = (15y + 4)^2
\][/tex]
### 19. Factor the difference of cubes: [tex]\(64x^3 - 125\)[/tex]
1. Recognize the difference of cubes:
We have [tex]\(64x^3 = (4x)^3\)[/tex] and [tex]\(125 = 5^3\)[/tex].
2. Use the formula for the difference of cubes:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
In this case, [tex]\(a = 4x\)[/tex] and [tex]\(b = 5\)[/tex].
3. Factor the expression:
[tex]\[
64x^3 - 125 = (4x - 5)((4x)^2 + (4x)(5) + 5^2)
\][/tex]
4. Simplify the second factor:
[tex]\[
(4x)^2 + 20x + 25 = 16x^2 + 20x + 25
\][/tex]
5. Final factorization:
[tex]\[
(4x - 5)(16x^2 + 20x + 25)
\][/tex]
### 26. Factor the quadratic polynomial: [tex]\(20w^2 - 47w + 24\)[/tex]
1. Find two numbers that multiply to [tex]\(20 \times 24 = 480\)[/tex] and add to [tex]\(-47\)[/tex]:
The numbers are [tex]\(-40\)[/tex] and [tex]\(-7\)[/tex].
2. Split the middle term using these numbers:
[tex]\[
20w^2 - 40w - 7w + 24
\][/tex]
3. Factor by grouping:
[tex]\[
20w^2 - 40w - 7w + 24 = (20w^2 - 40w) + (-7w + 24)
\][/tex]
[tex]\[
= 20w(w - 2) - 7(w - 2)
\][/tex]
4. Factor out the common term (w - 2):
[tex]\[
= (20w - 7)(w - 2)
\][/tex]
5. Final factorization:
[tex]\[
= (4w - 3)(5w - 8)
\][/tex]
### 32. Factor the quadratic polynomial: [tex]\(90v^2 - 181v + 90\)[/tex]
1. Find two numbers that multiply to [tex]\(90 \times 90 = 8100\)[/tex] and add to [tex]\(-181\)[/tex]:
The numbers are [tex]\(-90\)[/tex] and [tex]\(-91\)[/tex].
2. Split the middle term using these numbers:
[tex]\[
90v^2 - 90v - 91v + 90
\][/tex]
3. Factor by grouping:
[tex]\[
90v^2 - 90v - 91v + 90 = (90v^2 - 90v) + (-91v + 90)
\][/tex]
[tex]\[
= 90v(v - 1) - 91(v - 1)
\][/tex]
4. Factor out the common term (v - 1):
[tex]\[
= (90v - 91)(v - 1)
\][/tex]
5. Final factorization:
[tex]\[
= (9v - 10)(10v - 9)
\][/tex]
### 36. Completely factor the polynomial: [tex]\(x^3 + x^2 - 20x\)[/tex]
1. Factor out the greatest common factor, [tex]\(x\)[/tex]:
[tex]\[
x(x^2 + x - 20)
\][/tex]
2. Factor the quadratic [tex]\(x^2 + x - 20\)[/tex]:
- Find two numbers that multiply to [tex]\(-20\)[/tex] and add to [tex]\(1\)[/tex]: The numbers are [tex]\(5\)[/tex] and [tex]\(-4\)[/tex].
3. Factor the quadratic:
[tex]\[
x^2 + x - 20 = (x + 5)(x - 4)
\][/tex]
4. Final factorization:
[tex]\[
x(x + 5)(x - 4)
\][/tex]
### 40. Completely factor the polynomial: [tex]\(2x^3 - x^2 - 8x + 4\)[/tex]
1. Look for common factor in pairs:
[tex]\[
(2x^3 - x^2) + (-8x + 4)
\][/tex]
2. Factor each pair:
[tex]\[
x^2(2x - 1) - 4(2x - 1)
\][/tex]
3. Combine the factored terms:
[tex]\[
= (x^2 - 4)(2x - 1)
\][/tex]
4. Factor [tex]\(x^2 - 4\)[/tex] further using difference of squares:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
5. Final factorization:
[tex]\[
(x - 2)(x + 2)(2x - 1)
\][/tex]
These steps cover the detailed factorization process for each selected problem. If you have any more questions or need further clarification, feel free to ask!
