Answer :
Let's go through the exercise step-by-step to solve these algebraic questions systematically.
### 1. Common Monomial Factor (CMF)
Example a:
[tex]\[ 5a + 10 = 5(a + 2) \][/tex]
b:
[tex]\[ 4x^2y^2 - 12xy = 4xy(xy - 3) \][/tex]
c:
[tex]\[ 10z^3 + 15x \][/tex]
To find the CMF in [tex]\(10z^3 + 15x\)[/tex]:
1. Find the greatest common factor of the coefficients 10 and 15, which is 5.
2. Both terms share the factor 5.
So, we factor out 5:
[tex]\[ 10z^3 + 15x = 5(2z^3 + 3x) \][/tex]
d:
[tex]\[ 8p^2 + 14p - 4 \][/tex]
To find the CMF in [tex]\(8p^2 + 14p - 4\)[/tex]:
1. 8, 14, and 4 are divisible by 2, which is their greatest common factor.
So, we factor out 2:
[tex]\[ 8p^2 + 14p - 4 = 2(4p^2 + 7p - 2) \][/tex]
### 2. Difference of Two Squares
Example a:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
b:
[tex]\[ 100r^4 - 25s^2 = (10r^2 + 5s)(10r^2 - 5s) \][/tex]
c:
[tex]\[ a^2 - 49 \][/tex]
Here, 49 is 7 squared:
[tex]\[ a^2 - 49 = (a + 7)(a - 7) \][/tex]
d:
[tex]\[ 121x^4 - 9 \][/tex]
Here, 121 is 11 squared and 9 is 3 squared:
[tex]\[ 121x^4 - 9 = (11x^2 + 3)(11x^2 - 3) \][/tex]
### 3. Sum of Two Cubes
Example a:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
b:
[tex]\[ 27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1) \][/tex]
c:
[tex]\[ a^3 - 125 \][/tex]
Here, 125 is 5 cubed:
[tex]\[ a^3 - 125 = (a - 5)(a^2 + 5a + 25) \][/tex]
d:
[tex]\[ 8y^3 - 27 \][/tex]
Here, 8 is 2 cubed and 27 is 3 cubed:
[tex]\[ 8y^3 - 27 = (2y - 3)(4y^2 + 6y + 9) \][/tex]
### 5. Factoring Quadratic Trinomials (where [tex]\(a=1\)[/tex])
Example a:
[tex]\[ a^2 + 11a + 18 = (a + 9)(a + 2) \][/tex]
b:
[tex]\[ k^2 - 5k + 6 = (k - 3)(k - 2) \][/tex]
c:
[tex]\[ x^2 + 8x + 15 \][/tex]
Finding two numbers that multiply to 15 and add to 8:
[tex]\[ x^2 + 8x + 15 = (x + 3)(x + 5) \][/tex]
d:
[tex]\[ m^2 - 9m + 14 \][/tex]
Finding two numbers that multiply to 14 and add to -9:
[tex]\[ m^2 - 9m + 14 = (m - 2)(m - 7) \][/tex]
### 5. Give the factors of [tex]\(12x^2 + 6x\)[/tex]
A:
[tex]\[ 6(2x^2 + x) \][/tex]
B:
[tex]\[ 12(2) \text{ but wasn't fully written in the question; assume there was an error or truncation.} \][/tex]
### 6. The Product of Sum and Difference
Here, product of sum and difference is typically:
[tex]\[ (a + b)(a - b) = a^2 - b^2 \][/tex]
### 7. What are the factors of [tex]\(\frac{4}{y}x^2 - \cdots\)[/tex]? (Incomplete question)
We would typically look for common factors or employ methods like the difference of squares, but since the question is incomplete, we can't solve it fully.
### 8. Area of a Square is [tex]\(9x^2\)[/tex], find the side
The side of the square given area [tex]\(9x^2\)[/tex]:
[tex]\[ \text{Side} = \sqrt{9x^2} = 3x \][/tex]
### 9. One of the factors of [tex]\(2x^2\)[/tex]
It's generally:
[tex]\[ x \text{ or a constant factor which wasn't fully mentioned.} \][/tex]
### 10. Which of the following? (Multiple choices weren't provided)
### 11. Find the missing term
For completeness, it asks to find the [tex]\(n\)[/tex]-th term of a common mathematical sequence or expansion, but again it’s not provided fully in the initial question.
Overall, these steps would help you systematically address and solve algebraic problems involving common factors, difference of squares, sum of cubes, and factoring quadratic trinomials.
### 1. Common Monomial Factor (CMF)
Example a:
[tex]\[ 5a + 10 = 5(a + 2) \][/tex]
b:
[tex]\[ 4x^2y^2 - 12xy = 4xy(xy - 3) \][/tex]
c:
[tex]\[ 10z^3 + 15x \][/tex]
To find the CMF in [tex]\(10z^3 + 15x\)[/tex]:
1. Find the greatest common factor of the coefficients 10 and 15, which is 5.
2. Both terms share the factor 5.
So, we factor out 5:
[tex]\[ 10z^3 + 15x = 5(2z^3 + 3x) \][/tex]
d:
[tex]\[ 8p^2 + 14p - 4 \][/tex]
To find the CMF in [tex]\(8p^2 + 14p - 4\)[/tex]:
1. 8, 14, and 4 are divisible by 2, which is their greatest common factor.
So, we factor out 2:
[tex]\[ 8p^2 + 14p - 4 = 2(4p^2 + 7p - 2) \][/tex]
### 2. Difference of Two Squares
Example a:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
b:
[tex]\[ 100r^4 - 25s^2 = (10r^2 + 5s)(10r^2 - 5s) \][/tex]
c:
[tex]\[ a^2 - 49 \][/tex]
Here, 49 is 7 squared:
[tex]\[ a^2 - 49 = (a + 7)(a - 7) \][/tex]
d:
[tex]\[ 121x^4 - 9 \][/tex]
Here, 121 is 11 squared and 9 is 3 squared:
[tex]\[ 121x^4 - 9 = (11x^2 + 3)(11x^2 - 3) \][/tex]
### 3. Sum of Two Cubes
Example a:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
b:
[tex]\[ 27x^3 - 1 = (3x - 1)(9x^2 + 3x + 1) \][/tex]
c:
[tex]\[ a^3 - 125 \][/tex]
Here, 125 is 5 cubed:
[tex]\[ a^3 - 125 = (a - 5)(a^2 + 5a + 25) \][/tex]
d:
[tex]\[ 8y^3 - 27 \][/tex]
Here, 8 is 2 cubed and 27 is 3 cubed:
[tex]\[ 8y^3 - 27 = (2y - 3)(4y^2 + 6y + 9) \][/tex]
### 5. Factoring Quadratic Trinomials (where [tex]\(a=1\)[/tex])
Example a:
[tex]\[ a^2 + 11a + 18 = (a + 9)(a + 2) \][/tex]
b:
[tex]\[ k^2 - 5k + 6 = (k - 3)(k - 2) \][/tex]
c:
[tex]\[ x^2 + 8x + 15 \][/tex]
Finding two numbers that multiply to 15 and add to 8:
[tex]\[ x^2 + 8x + 15 = (x + 3)(x + 5) \][/tex]
d:
[tex]\[ m^2 - 9m + 14 \][/tex]
Finding two numbers that multiply to 14 and add to -9:
[tex]\[ m^2 - 9m + 14 = (m - 2)(m - 7) \][/tex]
### 5. Give the factors of [tex]\(12x^2 + 6x\)[/tex]
A:
[tex]\[ 6(2x^2 + x) \][/tex]
B:
[tex]\[ 12(2) \text{ but wasn't fully written in the question; assume there was an error or truncation.} \][/tex]
### 6. The Product of Sum and Difference
Here, product of sum and difference is typically:
[tex]\[ (a + b)(a - b) = a^2 - b^2 \][/tex]
### 7. What are the factors of [tex]\(\frac{4}{y}x^2 - \cdots\)[/tex]? (Incomplete question)
We would typically look for common factors or employ methods like the difference of squares, but since the question is incomplete, we can't solve it fully.
### 8. Area of a Square is [tex]\(9x^2\)[/tex], find the side
The side of the square given area [tex]\(9x^2\)[/tex]:
[tex]\[ \text{Side} = \sqrt{9x^2} = 3x \][/tex]
### 9. One of the factors of [tex]\(2x^2\)[/tex]
It's generally:
[tex]\[ x \text{ or a constant factor which wasn't fully mentioned.} \][/tex]
### 10. Which of the following? (Multiple choices weren't provided)
### 11. Find the missing term
For completeness, it asks to find the [tex]\(n\)[/tex]-th term of a common mathematical sequence or expansion, but again it’s not provided fully in the initial question.
Overall, these steps would help you systematically address and solve algebraic problems involving common factors, difference of squares, sum of cubes, and factoring quadratic trinomials.
