For , which expression is equivalent to ?

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SAT Equivalent Expressions Question 90 | Free SAT Prep | Aniko

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Question 90·Hard·Equivalent Expressions

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$P(z)=z^4-5z^2+4$

$Q(z)=z^2-3z+2$

For $z>2$ , which expression is equivalent to $\dfrac{P(z)}{Q(z)}$ ?

For rational expressions with polynomial numerator and denominator, first factor both completely instead of doing long division. Look for patterns (like treating $z^4 - 5z^2 + 4$ as a quadratic in $z^2$ and using difference of squares). Once factored, cancel any common factors, making sure those factors are not zero on the given domain (here, $z>2$ ensures $z-1$ and $z-2$ are nonzero). The resulting simplified expression can then be matched quickly to the answer choices, sometimes confirmed by plugging in a simple allowed value like $z=3$ to check your work.

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Think about factoring

Instead of trying to divide the polynomials directly, think about factoring the numerator and the denominator. Can you write each as a product of simpler expressions?

Treat $z^4 - 5z^2 + 4$ as a quadratic

Notice that $z^4 - 5z^2 + 4$ can be seen as $(z^2)^2 - 5(z^2) + 4$ . How do you factor a quadratic of the form $x^2 - 5x + 4$ ?

Use the condition $z>2$ when canceling

After factoring, you should see common factors in the numerator and denominator. Remember that you can only cancel a factor if it is not zero; how does $z>2$ help you justify the cancellation?

Compare to the answer choices

Once you have your simplified factored expression, look for the answer choice that matches it exactly (you may need to expand a product mentally to compare).

Desmos Guide

Enter the original expression

In one expression line, type (x^4 - 5x^2 + 4)/(x^2 - 3x + 2) (use x in place of $z$ ). You can either look at the graph for $x>2$ or evaluate it at a specific $x$ -value like $x=3$ using the calculator line.

Compare with each answer choice

For each choice, enter its expression in a new line: x+3, (x+1)(x+2), (x+2)(x-1), and x^2+3x-2. Either:

Graph them and compare which graph exactly overlaps the original for $x>2$ , or
Evaluate each at the same $x$ -value (such as $x=3$ ) and see which one gives the same numeric result as the original expression. The matching one is the correct choice.

Step-by-step Explanation

Recognize the goal: simplify a rational expression

You are given

\frac{P(z)}{Q(z)} = \frac{z^4 - 5z^2 + 4}{z^2 - 3z + 2}

and asked which choice is equivalent for $z>2$ . That means you should algebraically simplify the fraction (like reducing a numeric fraction) by factoring and canceling, being careful about when cancellation is allowed.

Factor the denominator $Q(z)$

Factor the quadratic in the denominator:

Q(z) = z^2 - 3z + 2.

Look for two numbers that multiply to $2$ and add to $-3$ . Those numbers are $-1$ and $-2$ , so

Q(z) = (z - 1)(z - 2).

Factor the numerator $P(z)$

Rewrite the numerator as a quadratic in $z^2$ :

P(z) = z^4 - 5z^2 + 4 = (z^2)^2 - 5(z^2) + 4.

Now factor this like a quadratic. You need two numbers that multiply to $4$ and add to $-5$ : they are $-1$ and $-4$ .

P(z) = (z^2 - 1)(z^2 - 4).

Each of these is a difference of squares:

\begin{aligned} P(z) &= (z^2 - 1)(z^2 - 4) \\ &= (z - 1)(z + 1)(z - 2)(z + 2). \end{aligned}

Form the fraction and cancel common factors (using $z>2$ )

Now write the fraction with all factors visible:

\frac{P(z)}{Q(z)} = \frac{(z - 1)(z + 1)(z - 2)(z + 2)}{(z - 1)(z - 2)}.

For $z>2$ , $z-1$ and $z-2$ are not zero, so it is legal to cancel these common factors from numerator and denominator:

\frac{(z - 1)(z + 1)(z - 2)(z + 2)}{(z - 1)(z - 2)} = (z + 1)(z + 2).

So $\dfrac{P(z)}{Q(z)}$ simplifies to $(z+1)(z+2)$ for $z>2$ , which corresponds to choice B.

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