Question 56·SAT Desmos Regression Practice·Advanced Math

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A materials scientist models the stress on a test rod after $x$ minutes with a polynomial $S(x)$ . One report gives

S(x)=(x+1)^3+p(x-2)^3+qx+r,

and another report gives

S(x)=5x^3-21x^2+40x-10.

If the two expressions for $S(x)$ are equivalent for all real $x$ , which choice gives the value of $p+q+r$ ?

The phrase equivalent for all real $x$ signals a polynomial identity: the two reports describe the same polynomial, not two curves that happen to meet once. With several unknown constants, use an identity regression in Desmos to fit them at multiple inputs, then add the values the question asks for. One matching point alone isn't enough.

Hints

Hint 1
An identity is an equation that holds at every input. A single point where two graphs meet could be a coincidence, so use several different inputs to determine the unknown constants.
Hint 2
For an identity regression, give Desmos a list such as $x_1=[1...5]$ . Replace each $x$ with $x_1$ and use $\sim$ instead of $=$ ; Desmos reports the fitted constants under PARAMETERS.
Hint 3
The target is $p+q+r$ , not any one constant. After Desmos finds the three values, type their sum on a new line so you don't stop at an intermediate result.

Step-by-step

Approach 1: Fit the identity in Desmos
Step 1
Use the meaning of an identity
The reports are equivalent for all real $x$ , so their difference must be zero everywhere. Both are cubics. A nonzero cubic can be zero at no more than three distinct inputs, so four matching inputs force two cubics to be identical. That makes a list of five inputs enough to fit and check the unknown constants.
Step 2
Fit the three constants
Type $x_1=[1...5]$ to give Desmos five inputs. Then type the two reports as shown below, replacing $x$ with $x_1$ and $=$ with $\sim$ ; Desmos uses $\sim$ to fit the unknown letters rather than graph the equation. Under PARAMETERS, it shows $p=4$ , $q=-11$ , and $r=21$ .
Step 3
Calculate the requested sum
The question asks for $p+q+r$ , so add that as a new Desmos line. It displays $14$ . The sum of the three constants is $14$ . Choice B.

Approach 2: Match one coefficient, then choose an input
Step 1
Match the cubic terms
In an identity, terms with the same power of $x$ must have the same coefficient. The $x^3$ terms on the first report are $x^3$ and $px^3$ ; $qx+r$ adds none. Match their combined coefficient to the $5x^3$ on the second report:
$1+p=5$
Step 2
Find $p$
Subtract $1$ from both sides:
$p=4$
Don't take $p=5$ : $(x+1)^3$ already contributes $1x^3$ .
Step 3
Choose an input that keeps $q$ and $r$
At $x=1$ , the term $qx$ becomes $q$ and the constant stays $r$ . Put $1$ into the first report:
$S(1)=(1+1)^3+p(1-2)^3+q(1)+r$
Step 4
Show why this input gives the sum
Evaluate the powers:
$S(1)=8-p+q+r$
Replace $p$ with $4$ :
$S(1)=8-4+q+r$
Subtract $4$ from $8$ :
$S(1)=4+q+r=p+q+r$
So evaluating $S(1)$ gives the requested sum without finding $q$ and $r$ separately.
Step 5
Evaluate the second report at $1$
Define $S(x)$ using the second report, then type $S(1)$ . Desmos displays $14$ . Since $S(1)=p+q+r$ , the requested sum is $14$ . Choice B.

Question 56·SAT Desmos Regression Practice·Advanced Math

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