Answer :
Let's go through the given expressions and factor them step by step:
### Part a
Expression: \( x^2 + 10x + 21 \)
We need to find two numbers that multiply to \( 21 \) and add up to \( 10 \). Considering the factors of 21, we have:
[tex]\[ 3 \cdot 7 = 21 \quad \text{and} \quad 3 + 7 = 10 \][/tex]
Thus, the factors are:
[tex]\[ x^2 + 10x + 21 = (x + 3)(x + 7) \][/tex]
### Part b
Expression: \( x^2 - 13x + 12 \)
We need to find two numbers that multiply to \( 12 \) and add up to \( -13 \). Considering the factors of 12, we have:
[tex]\[ -1 \cdot -12 = 12 \quad \text{and} \quad -1 + (-12) = -13 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 13x + 12 = (x - 12)(x - 1) \][/tex]
### Part c
Expression: \( x^2 - 6x - 16 \)
We need to find two numbers that multiply to \( -16 \) and add up to \( -6 \). Considering the factors of -16, we have:
[tex]\[ -8 \cdot 2 = -16 \quad \text{and} \quad -8 + 2 = -6 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 6x - 16 = (x - 8)(x + 2) \][/tex]
### Part d
Expression: \( x^2 - 11x - 26 \)
We need to find two numbers that multiply to \( -26 \) and add up to \( -11 \). Considering the factors of -26, we have:
[tex]\[ -13 \cdot 2 = -26 \quad \text{and} \quad -13 + 2 = -11 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 11x - 26 = (x - 13)(x + 2) \][/tex]
### Part e
Expression: \( x^2 - 13x + 42 \)
We need to find two numbers that multiply to \( 42 \) and add up to \( -13 \). Considering the factors of 42, we have:
[tex]\[ -7 \cdot -6 = 42 \quad \text{and} \quad -7 + (-6) = -13 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 13x + 42 = (x - 7)(x - 6) \][/tex]
### Part f
Expression: \( x^2 - 15x + 54 \)
We need to find two numbers that multiply to \( 54 \) and add up to \( -15 \). Considering the factors of 54, we have:
[tex]\[ -9 \cdot -6 = 54 \quad \text{and} \quad -9 + (-6) = -15 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 15x + 54 = (x - 9)(x - 6) \][/tex]
### Part g
Expression: \( x^2 + 20x + 99 \)
We need to find two numbers that multiply to \( 99 \) and add up to \( 20 \). Considering the factors of 99, we have:
[tex]\[ 9 \cdot 11 = 99 \quad \text{and} \quad 9 + 11 = 20 \][/tex]
Thus, the factors are:
[tex]\[ x^2 + 20x + 99 = (x + 9)(x + 11) \][/tex]
### Part h
Expression: \( x^2 - 3xy - 18y^2 \)
We need to find two numbers that multiply to \( -18y^2 \) and add up to \( -3y \). Considering the factors, we have:
[tex]\[ -6y \cdot 3y = -18y^2 \quad \text{and} \quad -6y + 3y = -3y \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 3xy - 18y^2 = (x - 6y)(x + 3y) \][/tex]
### Part i
Expression: \( x^2 + 19xy + 48y^2 \)
We need to find two numbers that multiply to \( 48y^2 \) and add up to \( 19y \). Considering the factors, we have:
[tex]\[ 3y \cdot 16y = 48y^2 \quad \text{and} \quad 3y + 16y = 19y \][/tex]
Thus, the factors are:
[tex]\[ x^2 + 19xy + 48y^2 = (x + 3y)(x + 16y) \][/tex]
### Part ј
Expression: \( 6x^2 + 24x + 18 \)
First, we factor out the greatest common factor (GCF), which is 6:
[tex]\[ 6x^2 + 24x + 18 = 6(x^2 + 4x + 3) \][/tex]
Now, factor \( x^2 + 4x + 3 \):
[tex]\[ x^2 + 4x + 3 = (x + 3)(x + 1) \][/tex]
Thus, the factors are:
[tex]\[ 6x^2 + 24x + 18 = 6(x + 3)(x + 1) \][/tex]
### Part k
Expression: \( 3x^2 - 18x + 48 \)
First, we factor out the greatest common factor (GCF), which is 3:
[tex]\[ 3x^2 - 18x + 48 = 3(x^2 - 6x + 16) \][/tex]
The expression \( x^2 - 6x + 16 \) does not factor nicely over the integers, so it remains as it is.
Thus, the factors are:
[tex]\[ 3x^2 - 18x + 48 = 3(x^2 - 6x + 16) \][/tex]
### Part l
Expression: \( 7x^2 - 35x + 42 \)
First, we factor out the greatest common factor (GCF), which is 7:
[tex]\[ 7x^2 - 35x + 42 = 7(x^2 - 5x + 6) \][/tex]
Now, factor \( x^2 - 5x + 6 \):
[tex]\[ x^2 - 5x + 6 = (x - 3)(x - 2) \][/tex]
Thus, the factors are:
[tex]\[ 7x^2 - 35x + 42 = 7(x - 3)(x - 2) \][/tex]
### Part a
Expression: \( x^2 + 10x + 21 \)
We need to find two numbers that multiply to \( 21 \) and add up to \( 10 \). Considering the factors of 21, we have:
[tex]\[ 3 \cdot 7 = 21 \quad \text{and} \quad 3 + 7 = 10 \][/tex]
Thus, the factors are:
[tex]\[ x^2 + 10x + 21 = (x + 3)(x + 7) \][/tex]
### Part b
Expression: \( x^2 - 13x + 12 \)
We need to find two numbers that multiply to \( 12 \) and add up to \( -13 \). Considering the factors of 12, we have:
[tex]\[ -1 \cdot -12 = 12 \quad \text{and} \quad -1 + (-12) = -13 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 13x + 12 = (x - 12)(x - 1) \][/tex]
### Part c
Expression: \( x^2 - 6x - 16 \)
We need to find two numbers that multiply to \( -16 \) and add up to \( -6 \). Considering the factors of -16, we have:
[tex]\[ -8 \cdot 2 = -16 \quad \text{and} \quad -8 + 2 = -6 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 6x - 16 = (x - 8)(x + 2) \][/tex]
### Part d
Expression: \( x^2 - 11x - 26 \)
We need to find two numbers that multiply to \( -26 \) and add up to \( -11 \). Considering the factors of -26, we have:
[tex]\[ -13 \cdot 2 = -26 \quad \text{and} \quad -13 + 2 = -11 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 11x - 26 = (x - 13)(x + 2) \][/tex]
### Part e
Expression: \( x^2 - 13x + 42 \)
We need to find two numbers that multiply to \( 42 \) and add up to \( -13 \). Considering the factors of 42, we have:
[tex]\[ -7 \cdot -6 = 42 \quad \text{and} \quad -7 + (-6) = -13 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 13x + 42 = (x - 7)(x - 6) \][/tex]
### Part f
Expression: \( x^2 - 15x + 54 \)
We need to find two numbers that multiply to \( 54 \) and add up to \( -15 \). Considering the factors of 54, we have:
[tex]\[ -9 \cdot -6 = 54 \quad \text{and} \quad -9 + (-6) = -15 \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 15x + 54 = (x - 9)(x - 6) \][/tex]
### Part g
Expression: \( x^2 + 20x + 99 \)
We need to find two numbers that multiply to \( 99 \) and add up to \( 20 \). Considering the factors of 99, we have:
[tex]\[ 9 \cdot 11 = 99 \quad \text{and} \quad 9 + 11 = 20 \][/tex]
Thus, the factors are:
[tex]\[ x^2 + 20x + 99 = (x + 9)(x + 11) \][/tex]
### Part h
Expression: \( x^2 - 3xy - 18y^2 \)
We need to find two numbers that multiply to \( -18y^2 \) and add up to \( -3y \). Considering the factors, we have:
[tex]\[ -6y \cdot 3y = -18y^2 \quad \text{and} \quad -6y + 3y = -3y \][/tex]
Thus, the factors are:
[tex]\[ x^2 - 3xy - 18y^2 = (x - 6y)(x + 3y) \][/tex]
### Part i
Expression: \( x^2 + 19xy + 48y^2 \)
We need to find two numbers that multiply to \( 48y^2 \) and add up to \( 19y \). Considering the factors, we have:
[tex]\[ 3y \cdot 16y = 48y^2 \quad \text{and} \quad 3y + 16y = 19y \][/tex]
Thus, the factors are:
[tex]\[ x^2 + 19xy + 48y^2 = (x + 3y)(x + 16y) \][/tex]
### Part ј
Expression: \( 6x^2 + 24x + 18 \)
First, we factor out the greatest common factor (GCF), which is 6:
[tex]\[ 6x^2 + 24x + 18 = 6(x^2 + 4x + 3) \][/tex]
Now, factor \( x^2 + 4x + 3 \):
[tex]\[ x^2 + 4x + 3 = (x + 3)(x + 1) \][/tex]
Thus, the factors are:
[tex]\[ 6x^2 + 24x + 18 = 6(x + 3)(x + 1) \][/tex]
### Part k
Expression: \( 3x^2 - 18x + 48 \)
First, we factor out the greatest common factor (GCF), which is 3:
[tex]\[ 3x^2 - 18x + 48 = 3(x^2 - 6x + 16) \][/tex]
The expression \( x^2 - 6x + 16 \) does not factor nicely over the integers, so it remains as it is.
Thus, the factors are:
[tex]\[ 3x^2 - 18x + 48 = 3(x^2 - 6x + 16) \][/tex]
### Part l
Expression: \( 7x^2 - 35x + 42 \)
First, we factor out the greatest common factor (GCF), which is 7:
[tex]\[ 7x^2 - 35x + 42 = 7(x^2 - 5x + 6) \][/tex]
Now, factor \( x^2 - 5x + 6 \):
[tex]\[ x^2 - 5x + 6 = (x - 3)(x - 2) \][/tex]
Thus, the factors are:
[tex]\[ 7x^2 - 35x + 42 = 7(x - 3)(x - 2) \][/tex]
