Question 66·SAT Desmos Regression Practice·Advanced Math

Type $x_1=[1,2,3,5,6]$; Desmos displays the five numbers. They avoid $0$ and $4$, which make the given denominator zero. Several allowed inputs give you enough comparisons to determine four unknown constants.

Type $\frac{x_1^4-16}{x_1^2-4x_1}\sim x_1^2+px_1+q+\frac{a}{x_1}+\frac{b}{x_1-4}$. The $x_1$ uses your input list, and $\sim$ tells Desmos to fit the constants. Under PARAMETERS, Desmos shows $p=4$, $q=16$, $a=4$, and $b=60$. Use several allowed inputs to find the constants in an identity.

Type $p+q+a+b$; Desmos prints $84$. Include both fraction numerators, $a$ and $b$, rather than stopping at the polynomial coefficients. So $p+q+a+b=84$. Choice C.

To divide by $x^2-4x$, first cancel $x^4$ by subtracting $x^2$ times the denominator:

The result is the remainder, the part still to divide.

The remainder now starts with $4x^3$. Cancel it by subtracting $4x$ times the same denominator:

Cancel $16x^2$ by subtracting $16$ times the denominator:

This leftover has a lower power than the denominator, so division stops.

The multiples you used, $x^2$, $4x$, and $16$, form the quotient, or whole-polynomial part. Put the remainder over the original denominator:

So the polynomial coefficients are $p=4$ and $q=16$.

Each fraction needs the factor missing from its denominator. Use the common denominator $x(x-4)$:

The two remainder fractions have the same denominator and agree for every allowed $x$, so their numerators must agree:

Distribute $a$ and collect the $x$ terms so you can match their coefficients, the numbers multiplying each power:

The $64$ belongs to $a+b$, not to $a$ alone.

Match the terms without $x$:

Divide by $-4$:

The constant term determines $a$ directly.

Match the coefficients of $x$:

Subtract $a$:

Substitute $a=4$:

So $64$ is the sum of the two fraction numerators.

Type $4+16+4+60$ in Desmos; it prints $84$. Those are $p$, $q$, $a$, and $b$, respectively, so $p+q+a+b=84$. Choice C.