Question 65·SAT Desmos Regression Practice·Advanced Math

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For all real numbers $x$ such that $x \ne -4$ , $x \ne -1$ , and $x \ne 4$ , the expression

\frac{ax^3+cx+d}{(x^2-16)(x+1)}

is equivalent to

\frac{5}{x-4}+\frac{m}{x+4}+\frac{2x-1}{x+1}.

In these expressions, $a$ , $c$ , $d$ , and $m$ are constants. Which choice is the value of $a+c$ ?

Two rational expressions that agree at every allowed input form an identity: the same equation holds wherever the denominators aren't zero. Use a Desmos regression with several allowed inputs to find the unknown constants. Clearing the common denominator explains why those constants are fixed. Be careful with a coefficient sum: evaluating the numerator at $x=1$ also includes its constant term.

Hints

Hint 1
The words equivalent for all allowed inputs give you an identity, an equation that holds wherever both expressions are defined. Write the two given expressions on opposite sides of an equals sign.
Hint 2
One input gives only one equation, but there are four unknown constants. Choose several distinct inputs that avoid $-4$ , $-1$ , and $4$ so Desmos can test the equality at all of them.
Hint 3
A regression finds constants that make an equation fit the inputs in a list. Replace every $x$ with $x_1$ and the equals sign with $\sim$ , then combine only the coefficients the question asks for.

Step-by-step

Approach 1: Fit the identity in Desmos
Step 1
Turn equivalence into an equation
Here, equivalent means the two expressions have equal values at every allowed $x$ . Write that claim as an equation: $\frac{ax^3+cx+d}{(x^2-16)(x+1)}=\frac{5}{x-4}+\frac{m}{x+4}+\frac{2x-1}{x+1}$ .
Step 2
Choose inputs the fractions allow
The domain is the set of inputs where an expression is defined. The problem excludes $-4$ , $-1$ , and $4$ , so choose $-3$ , $-2$ , $0$ , $1$ , and $2$ : none makes a denominator zero. Type $x_1=[-3,-2,0,1,2]$ . Desmos stores these five allowed inputs as one list for the next line.
Step 3
Find the constants with a regression
The equality must hold at all five chosen inputs, not only at one convenient value. Type the equation again with every $x$ changed to $x_1$ and $=$ changed to $\sim$ . This regression asks Desmos to find constants that make the sides agree across the list: $\frac{ax_1^3+cx_1+d}{(x_1^2-16)(x_1+1)}\sim\frac{5}{x_1-4}+\frac{m}{x_1+4}+\frac{2x_1-1}{x_1+1}$ . Under PARAMETERS, Desmos reports $a=2$ , $c=5$ , $d=52$ , and $m=-4$ .
Step 4
Add the requested coefficients
The requested sum uses $a$ and $c$ , not $m$ or the constant $d$ . Type $a+c$ below the regression; Desmos prints $7$ . So the sum of the coefficients of $x^3$ and $x$ is $7$ . Choice C.

Approach 2: See why the coefficients are fixed
Step 1
Find the common denominator
Instead of fitting inputs, use a difference of squares to factor the denominator: $x^2-16=(x-4)(x+4)$ . So the common denominator of every fraction is $D=(x-4)(x+4)(x+1)$ .
Step 2
Clear the denominators
The excluded inputs are exactly where $D=0$ , so $D$ is nonzero at every allowed input. Multiply the equation by $D$ . Each fraction's numerator gains the factors missing from its denominator: $ax^3+cx+d=5(x+4)(x+1)+m(x-4)(x+1)+(2x-1)(x^2-16)$ . Clearing the shared denominator turns equal fractions into equal polynomials.
Step 3
Use the missing squared term
The left polynomial has no $x^2$ term, so its $x^2$ coefficient is $0$ . The three products on the right contribute $5$ , $m$ , and $-1$ to that coefficient. Set their sum to $0$ : $5+m-1=0$ . Combine the numbers: $m+4=0$ . Subtract $4$ : $m=-4$ .
Step 4
Match the cubic term
Only $(2x-1)(x^2-16)$ makes an $x^3$ term: $2x\cdot x^2=2x^3$ . The left side's cubic coefficient is $a$ , so matching cubic coefficients gives $a=2$ .
Step 5
Match the linear term
For the $x$ coefficient, the three right-side products contribute $5(4+1)=25$ , $m(1-4)=-3m$ , and $2(-16)=-32$ . Match their sum to $c$ : $c=25-3m-32$ . Substitute $m=-4$ : $c=25-3(-4)-32$ . Multiply: $c=25+12-32$ . Combine the numbers: $c=5$ .
Step 6
Finish with the requested sum
Add only $a$ and $c$ : $a+c=2+5=7$ . So the sum of the coefficients of $x^3$ and $x$ is $7$ . Choice C.

Question 65·SAT Desmos Regression Practice·Advanced Math

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