QUESTION 1

[irfanakademisi]

QUESTION 1​

Which of the following is not a factor of the algebraic expression

[math]xy (x + 3)^2 - x^2 y (x + 3) + xy (x + 3)[/math]


[math]\text{A) } x \quad \text{B) } y \quad \text{C) } 3 \quad \text{D) } 4 \quad \text{E) } x+3[/math]


Solution:



[math]xy (x + 3)^2 - x^2 y (x + 3) + xy (x + 3)[/math]
Factoring out the greatest common factor [imath]xy(x + 3)[/imath]:

[math]= xy (x + 3) (x + 3 - x + 1)[/math]
[math]= 4xy (x + 3)[/math]
Evaluating the components reveals that 3 is not a factor of the expression.



[imath]\textbf{Correct Answer: C}[/imath]

QUESTION 2​

Which of the following choices represents a factor of the expression

[math]ab + a^2 b + a^3 b + a^4 b[/math]


[math]\text{A) } a^2b \quad \text{B) } ab^2\quad \text{C) } a+b \quad \text{D) } 1+a^2 \quad \text{E) } 1+a+a^2[/math]

Solution:​

First, extract the greatest common factor [imath]ab[/imath]:

[math]ab + a^2 b + a^3 b + a^4 b = ab (1 + a + a^2 + a^3)[/math]
Next, apply factoring by grouping within the parentheses:

[math]= ab ((1 + a) + a^2 (1 + a))[/math]
[math]= ab (1 + a) (1 + a^2)[/math]
Thus, [imath]1 + a^2[/imath] is a valid factor.



[imath]\textbf{Correct Answer: D}[/imath]

QUESTION 3​

Which of the following choices represents a factor of the expression

[math]3xy - 20ab - 15xb + 4ya[/math]


[math]\text{A) } x+a \quad \text{B) } y-5b\quad \text{C) } y+a \quad \text{D) } y-b \quad \text{E) } y+b[/math]

Solution:​

Rearrange the terms to group common variables together:

[math]3xy \;- \; 20ab\; -\; 15xb + 4ya[/math]
[math]= 3xy \;- \;15xb + 4ya \;- \;20ab[/math]
Factor by grouping the first two terms and the last two terms:

[math]= 3x (y \;-\; 5b) + 4a (y \;- \;5b)[/math]
[math]= (y \;- \;5b) (3x + 4a)[/math]
Thus, [imath]y - 5b[/imath] is a factor.



[imath]\textbf{Correct Answer: B}[/imath]

QUESTION 4​

Which of the following choices represents a factor of the expression

[math]2^x + 5^x + 6^x + (15)^x[/math]


[math]\text{A) } 1+ 3^x \quad \text{B) } 2^x + 3^x \quad \text{C) } 3^x+5^x \quad \text{D) } 1+ 5^x \quad \text{E) } 1+2^x[/math]

Solution:​

Rearrange the expression using the properties of exponents:

[math]2^x + 5^x + 6^x + (15)^x = 2^x + 6^x + 5^x + (15)^x[/math]
[math]= 2^x + 2^x \cdot 3^x + 5^x + 5^x \cdot 3^x[/math]
Factor by grouping the pairs:

[math]= 2^x (1 + 3^x) + 5^x (1 + 3^x)[/math]
[math]= (1 + 3^x) (2^x + 5^x)[/math]
Thus, [imath]1 + 3^x[/imath] is a factor.



[imath]\textbf{Correct Answer: A}[/imath]

QUESTION 5​

Given that [imath]a + b = 3[/imath] and [imath]b + c = 4[/imath], determine the numerical value of the expression:

[math]a^2 - bc + ab - ac[/math]


[math]\text{A) } 3 \quad \text{B) } -3 \quad \text{C) } 2 \quad \text{D) } -2 \quad \text{E) } 0[/math]

Solution:​

Rearrange and factor the expression by grouping:

[math]a^2 - bc + ab - ac = a^2 - ac + ab - bc[/math]
[math]= a (a - c) + b (a - c)[/math]
[math]= (a - c) (a + b)[/math]
We can find the value of [imath]a - c[/imath] by setting up a system of linear equations and subtracting the second equation from the first:

[math]\begin{array}{c} a + b = 3 \\ - (b + c = 4) \end{array}[/math]
[math]a - c = -1[/math]
Substituting the known values into the factored expression yields:

[math](a - c)(a + b) = (-1) \cdot 3 = -3[/math]


[imath]\textbf{Correct Answer: B}[/imath]




If anyone wants a clearer breakdown or more examples, I’m happy to share