Question 63·SAT Desmos Regression Practice·Advanced Math

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For some integer $p$ , the expression

\frac{x^3+px^2+12x}{x^3-x^2-12x}

is equivalent to

1+\frac{k}{x-4}

for all values of $x$ for which both expressions are defined.

Which choice gives the value of $k$ ?

The cue for all allowed values means these two rational expressions, fractions of polynomials, agree at every permitted input. First, find the inputs that make the original denominator zero. Then use a Desmos list regression at two legal inputs to find the two unknown constants. Two matching inputs alone wouldn't prove equivalence, but the question already promises it.

Hints

Hint 1
A denominator of $0$ makes a fraction undefined. Factor the original denominator before choosing inputs. Even if a factor cancels later, its zero is still forbidden in the original expression.
Hint 2
The problem gives an identity, an equality that holds at every allowed input. Since there are two unknown constants, try two different allowed inputs, such as $1$ and $2$ , to get information about both.
Hint 3
A Desmos regression fits unknown constants to the inputs you give it. Put $1$ and $2$ in a list named $x_1$ , then replace every $x$ in the given equality with $x_1$ and change $=$ to $\sim$ .

Step-by-step

Approach 1: Fit the two constants in Desmos
Step 1
Find the forbidden inputs
The denominator is the bottom of the fraction, and it can't be $0$ . Factor the original one:
$x^3-x^2-12x=x(x-4)(x+3).$
So $0$ , $4$ , and $-3$ are forbidden. In particular, don't use $-3$ merely because the expression on the right is defined there.
Step 2
Choose two allowed inputs
Type $x_1=[1,2]$ into Desmos; it shows the two entries in the list. Neither input is forbidden. Because equality is already promised for every allowed input, two legal inputs can pin down the two unknown constants. The next line will use both entries.
Step 3
Read the constant the question asks for
Replace each $x$ with $x_1$ and the equality sign with $\sim$ so Desmos fits $p$ and $k$ to both inputs. Type:
$\frac{x_1^3+px_1^2+12x_1}{x_1^3-x_1^2-12x_1}\sim1+\frac{k}{x_1-4}.$
Under PARAMETERS, Desmos shows $p=7$ and $k=8$ . The question asks for $k$ , not $p$ , so the value of $k$ is $8$ . Choice C.

Approach 2: Match the constant terms
Step 1
Cancel the shared factor
Factor $x$ out of the top and bottom, then cancel that whole factor. This works only where $x\ne0$ :
$\frac{x(x^2+px+12)}{x(x^2-x-12)}=\frac{x^2+px+12}{x^2-x-12}.$
Step 2
Expose the target denominator
Factor the remaining denominator to show the $x-4$ from the target form:
$x^2-x-12=(x-4)(x+3).$
So the given equality becomes
$\frac{x^2+px+12}{(x-4)(x+3)}=1+\frac{k}{x-4}.$
Step 3
Clear the denominators
For allowed $x$ , multiply both sides by $(x-4)(x+3)$ . The $1$ on the right contributes the entire product, while the fraction contributes $k(x+3)$ :
$x^2+px+12=(x-4)(x+3)+k(x+3).$
Step 4
Match the parts without x
This is an identity, so its constant terms, the parts without $x$ , must agree. The left contributes $12$ . On the right, $(x-4)(x+3)$ contributes $-12$ , and $k(x+3)$ contributes $3k$ :
$12=-12+3k.$
Step 5
Solve for k
Add $12$ to both sides:
$24=3k.$
Divide by $3$ :
$k=8.$
So the constant in the requested form is $8$ . Choice C.

Question 63·SAT Desmos Regression Practice·Advanced Math

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