Question 67·SAT Desmos Regression Practice·Advanced Math

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For all values of $x$ for which the expressions are defined,

\frac{3x^3-x^2+13x+4}{(x-2)^2(x+3)} = p+\frac{q}{x-2}+\frac{r}{(x-2)^2}+\frac{w}{x+3},

where $p$ , $q$ , $r$ , and $w$ are constants. What is the value of $q$ ?

Your answer

A partial-fraction decomposition writes one fraction as a sum of simpler fractions. The cue is “for all values of $x$ ” with unknown constants in those fractions. Clear the denominators, then use a Desmos regression at allowed inputs to find the constants. Don’t sample where an original denominator is zero, even though the cleared polynomial identity can be used there.

Hints

Hint 1
An excluded input makes a denominator zero. Both $x-2$ and $x+3$ appear below fractions here. Which two inputs must you leave out when testing the original equation?
Hint 2
Multiply every term by the whole original denominator. Each fraction then becomes a polynomial term, giving you an equation that is easier to fit at several inputs.
Hint 3
A polynomial of degree at most $3$ cannot have four distinct zeros unless it is zero everywhere. After clearing denominators, how many allowed inputs would establish that the two sides match?

Step-by-step

Approach 1: Clear denominators and fit the constants
Step 1
Find the inputs you cannot sample
A denominator cannot be $0$ . The factors $x-2$ and $x+3$ rule out $x=2$ and $x=-3$ , so neither input belongs in a regression on the original fractions.
Step 2
Turn the fraction equation into a polynomial identity
Multiply both sides by $(x-2)^2(x+3)$ , the common denominator. For instance, $q/(x-2)$ becomes $q(x-2)(x+3)$ :
$3x^3-x^2+13x+4=p(x-2)^2(x+3)+q(x-2)(x+3)+r(x+3)+w(x-2)^2.$
The polynomials agree at every allowed input, so they are identical. Only the denominators were cleared; the constants still have to be found.
Step 3
Use allowed inputs to find the requested constant
Name the original fraction $f$ . Type $f(x)=\frac{3x^3-x^2+13x+4}{(x-2)^2(x+3)}$ , then $x_1=[0,1,3,4,5]$ . All five inputs are allowed. Finally, type $f(x_1)\sim p+\frac{q}{x_1-2}+\frac{r}{(x_1-2)^2}+\frac{w}{x_1+3}$ . The $\sim$ tells Desmos to fit the constants to those values. A cubic difference that is zero at more than three distinct inputs is zero everywhere. Under PARAMETERS, Desmos shows $q=7$ . That is the coefficient of $1/(x-2)$ , so Grid in 7.

Approach 2: Isolate coefficients in the cleared identity
Step 1
Find the leading coefficient
In the cleared identity, only $p(x-2)^2(x+3)$ has an $x^3$ term. Its coefficient, meaning the number multiplying $x^3$ , must match the $3$ on the left:
$p=3.$
Step 2
Evaluate the numerator at a useful input
The cleared identity is a polynomial identity, so you may use $x=-3$ there even though the original fractions are undefined there. Name the numerator $N$ . Type $N(x)=3x^3-x^2+13x+4$ and $N(-3)$ ; Desmos shows $-125$ .
Step 3
Use that input to isolate $w$
At $x=-3$ , every term in the cleared identity except $w(x-2)^2$ contains $x+3$ and becomes $0$ . Substitute $N(-3)=-125$ :
$-125=w(-3-2)^2$ $-125=25w.$
Divide by $25$ :
$w=-5.$
Step 4
Identify the $x^2$ contributions
The coefficient of $x^2$ is the number multiplying $x^2$ . In $p(x-2)^2(x+3)$ , the $x^2$ pieces are $3x^2-4x^2=-x^2$ . Each of $(x-2)(x+3)$ and $(x-2)^2$ contributes $x^2$ . The $r(x+3)$ term contributes none, so the right-side coefficient is $-p+q+w$ .
Step 5
Match coefficients to find $q$
The left-side $x^2$ coefficient is $-1$ . Match it to the right-side coefficient:
$-1=-p+q+w.$
Substitute $p=3$ and $w=-5$ :
$-1=-3+q-5.$
Combine the numbers:
$-1=q-8.$
Add $8$ :
$q=7.$
So the coefficient asked for is $7$ . Grid in 7.

Question 67·SAT Desmos Regression Practice·Advanced Math

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