Question 58·SAT Desmos Regression Practice·Advanced Math

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For constants $p$ , $q$ , $r$ , and $s$ , the polynomial

4x^4+px^3+qx^2-72x+324

is equivalent to

(2x-6)^2(x^2+rx+s).

Which choice gives the value of $p+q$ ?

An identity says two polynomial forms agree for every value of $x$ . When several coefficients are missing, Desmos can use multiple inputs to fit them. Then evaluate the combination the question asks for. One matching input does not prove two expressions are equivalent. Also, a fitted constant may be only part of the requested sum.

Hints

Hint 1
An identity holds at every input. Give $x_1$ several values in Desmos, then use $\sim$ instead of $=$ to fit the missing constants rather than graph an equation.
Hint 2
A coefficient is the number multiplying a power of $x$ : $p$ multiplies $x^3$ , and $q$ multiplies $x^2$ . Once Desmos finds them, which two values give the requested sum?

Step-by-step

Approach 1: Fit the coefficients in Desmos
Step 1
Turn equivalence into an identity
The forms are equivalent, so they give the same value for every $x$ . Write that identity, an equation true at every input:
$4x^4+px^3+qx^2-72x+324=(2x-6)^2(x^2+rx+s).$
Step 2
Find the four missing constants
Type $x_1=[1...5]$ to supply five different inputs. Then type the identity with $x_1$ in place of $x$ and $\sim$ in place of $=$ . A regression fits constants so the two sides agree at those inputs. Under PARAMETERS, Desmos shows $p=-8$ , $q=-24$ , $r=4$ , and $s=9$ .
Step 3
Evaluate the requested sum
Type $p+q$ below the regression. Desmos prints $-32$ . That is the sum of the two coefficients, not either coefficient alone. Choice A.

Approach 2: Use coefficients, then choose a useful input
Step 1
Find the constant term
At $x=0$ , the factored form is $(-6)^2s=36s$ . It must equal the given constant term, $324$ :
$36s=324.$
Divide by $36$ :
$s=9.$
Step 2
Match the terms containing one $x$
In $(2x-6)^2$ , both cross products contribute an $x$ term:
$(2x)(-6)+(-6)(2x)=-24x.$
Multiplying by $x^2+rx+s$ , the $x$ terms are $-24sx$ and $36rx$ . Their coefficients must add to the given $-72$ :
$-24s+36r=-72.$
Step 3
Find $r$
Use $s=9$ . Substitute:
$-24(9)+36r=-72.$
Multiply:
$-216+36r=-72.$
Add $216$ :
$36r=144.$
Divide by $36$ :
$r=4.$
Step 4
Make $p+q$ appear
Because the forms agree at every input, choose $x=1$ . Both $x^3$ and $x^2$ then equal $1$ , so $px^3+qx^2$ becomes $p+q$ . Set the two values equal:
$4+p+q-72+324=(2-6)^2(1+r+s).$
Step 5
Calculate the coefficient sum
Substitute $r=4$ and $s=9$ , and square $2-6$ :
$4+p+q-72+324=16(1+4+9).$
Subtract the known terms on the left:
$p+q=16(1+4+9)-(4-72+324).$
Type the right side in Desmos. It prints $-32$ , so the requested coefficient sum is $-32$ . Choice A.

Question 58·SAT Desmos Regression Practice·Advanced Math

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