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Question 3·Hard·Equivalent Expressions

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Let $q = a - 3b$ . Which of the following is equivalent to the expression

\frac{a^2 - 6ab + 9b^2 - z^2}{a - 3b + z}

For this type of question, first use the given substitution (here, $q = a - 3b$ ) to rewrite any matching parts of the expression, especially the entire denominator and recognizable patterns in the numerator. Next, look for standard factoring patterns such as perfect square trinomials and differences of squares so you can factor the numerator cleanly in terms of the new variable. Finally, simplify the resulting fraction by canceling any common factors between the numerator and denominator; the remaining expression in terms of $q$ (and any other variables) corresponds to the correct answer choice. This process is usually faster and less error-prone than expanding or doing long algebra with the original variables.

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Use the definition of $q$

You are told that $q = a - 3b$ . Look at the denominator $a - 3b + z$ . How can you rewrite that using $q$ ?

Look closely at the first three terms of the numerator

Consider $a^2 - 6ab + 9b^2$ . Can you factor this part as a square of a binomial? How does that relate to $a - 3b$ and $q$ ?

Recognize a difference of squares and then simplify the fraction

Once you rewrite the numerator in terms of $q$ and $z$ , look for the pattern $A^2 - B^2 = (A - B)(A + B)$ . After factoring, see if any factor in the numerator matches the denominator so that the fraction can be simplified.

Desmos Guide

Define variables and the original expression

In Desmos, create sliders for $a$ , $b$ , and $z$ (for example, from $-5$ to $5$ ). Then enter the expression

E = \frac{a^2 - 6ab + 9b^2 - z^2}{a - 3b + z}

Make sure to avoid slider values where the denominator $a - 3b + z$ equals $0$ .

Define $q$ and each answer choice in terms of $a$ , $b$ , and $z$

Enter $q = a - 3b$ as a separate line. Then enter four more lines for the answer choices:

$A = q + z$
$B = (q - z)^2$
$C = (q + z)(q - z)$
$D = q - z$ . Desmos will show numerical values for $E$ , $A$ , $B$ , $C$ , and $D$ for each set of slider positions.

Compare values to see which expression matches

Move the sliders for $a$ , $b$ , and $z$ to several different non-problematic values (where the denominator is not $0$ ). For each set of values, compare the value of $E$ to $A$ , $B$ , $C$ , and $D$ . The correct choice is the one whose value always matches $E$ for every slider setting you try.

Step-by-step Explanation

Rewrite the denominator using $q$

We are given $q = a - 3b$ .

The denominator is $a - 3b + z$ , which can be rewritten using $q$ :

a - 3b + z = q + z.

So the denominator becomes $q + z$ .

Recognize and rewrite the squared part in the numerator

Look at the first three terms in the numerator:

a^2 - 6ab + 9b^2.

This is a perfect square trinomial:

a^2 - 6ab + 9b^2 = (a - 3b)^2.

Using $q = a - 3b$ , this becomes $q^2$ .

So the entire numerator is

a^2 - 6ab + 9b^2 - z^2 = (a - 3b)^2 - z^2 = q^2 - z^2.

Factor the numerator as a difference of squares

Now we have the numerator written as $q^2 - z^2$ .

This is a difference of squares, which factors as

q^2 - z^2 = (q - z)(q + z).

So the whole fraction becomes

\frac{a^2 - 6ab + 9b^2 - z^2}{a - 3b + z} = \frac{q^2 - z^2}{q + z} = \frac{(q - z)(q + z)}{q + z}.

Cancel the common factor to get the simplified expression

\frac{(q - z)(q + z)}{q + z},

$q + z$ is a common factor in the numerator and denominator (as long as $q + z \ne 0$ ), so it cancels:

\frac{(q - z)(q + z)}{q + z} = q - z.

So the expression is equivalent to $q - z$ , which corresponds to choice D.

Question 3·Hard·Equivalent Expressions

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation