Question

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for #1 to #4, match the polynomial in column a with an equivalent polynomial in column b. polynomials in column b may be used more than once or not at all.
column a

1. $dfrac{8xy}{2x}$
2. $dfrac{12x^2 - 6x}{3x}$
3. $(-2x + 1)(-2x)$
4. $dfrac{12xy - 6x}{3}$

column b
a $4xy - 2x$
b $4x^2 - 2x$
c $4y$
d $2x^2 - 2x$
e $4xy$
f $4x - 2$

Explanation:

Problem 1:

Step1: Simplify the fraction $\frac{8xy}{2x}$

We can divide the coefficients and the variables separately. For the coefficients, $8\div2 = 4$. For the variable $x$, $x\div x = 1$ (we can cancel out $x$ since $x
eq0$). The variable $y$ remains as it is. So $\frac{8xy}{2x}=4y$? Wait, no, wait. Wait, $8xy\div2x=(8\div2)\times(x\div x)\times y = 4\times1\times y = 4y$? But wait, looking at Column B, option E is $4xy$? Wait, no, maybe I made a mistake. Wait, no, let's re - check. Wait, the numerator is $8xy$ and the denominator is $2x$. So $\frac{8xy}{2x}=\frac{8}{2}\times\frac{x}{x}\times y = 4\times1\times y = 4y$? But in Column B, option C is $4y$. Wait, but maybe I misread the problem. Wait, no, maybe the original problem is $\frac{8xy}{2x}$. Let's do it again: $8xy\div2x=(8\div2)\times(x\div x)\times y = 4y$. So the equivalent polynomial in Column B is C? Wait, no, wait the Column B options: A: $4xy - 2x$, B: $4x^{2}-2x$, C: $4y$, D: $2x^{2}-2x$, E: $4xy$, F: $4x - 2$. So $\frac{8xy}{2x}=4y$, which is option C? Wait, but maybe I made a mistake. Wait, no, $x$ in the numerator and denominator cancels, $8\div2 = 4$, so we have $4y$. So the answer for problem 1 is C.

Step2: Match with Column B

After simplifying $\frac{8xy}{2x}$ to $4y$, we look at Column B and find that option C is $4y$.

Step1: Simplify the fraction $\frac{12x^{2}-6x}{3x}$

We can split the fraction into two parts: $\frac{12x^{2}}{3x}-\frac{6x}{3x}$. For the first part, $12x^{2}\div3x=(12\div3)\times(x^{2}\div x)=4x$. For the second part, $6x\div3x = 2$. So $\frac{12x^{2}-6x}{3x}=4x - 2$.

Step2: Match with Column B

Looking at Column B, option F is $4x - 2$.

Step1: Expand $(-2x + 1)(-2x)$

Using the distributive property (FOIL method), we multiply $-2x$ by $-2x$ and $1$ by $-2x$. So $(-2x)\times(-2x)+1\times(-2x)=4x^{2}-2x$.

Step2: Match with Column B

Looking at Column B, option B is $4x^{2}-2x$.

Answer:

C

Problem 2: