Thank you for visiting Instructions 1 Solve all problems but submit solutions only for problems 3 11 19 26 32 36 and 40 Exercises 1 7 Factor the difference. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
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### 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!
Thank you for reading the article Instructions 1 Solve all problems but submit solutions only for problems 3 11 19 26 32 36 and 40 Exercises 1 7 Factor the difference. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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