Question 57·SAT Desmos Regression Practice·Advanced Math

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The expression $(3x+2)^4-(3x-2)^4$ is equivalent to $ax^3+bx$ , where $a$ and $b$ are constants.

What is the value of $a-b$ ?

Your answer

An identity matches two expressions for every value of $x$ , not just one input. When an equivalent form has unknown coefficients, use a Desmos regression at several inputs to find them. Matching powers of $x$ must have matching coefficients. At $x=1$ , the coefficients in this form add rather than subtract, so keep the requested operation in view.

Hints

Hint 1
An identity is an equation that holds for every $x$ . Turn “equivalent to” into an equality between the given expression and $ax^3+bx$ ; you aren't solving for $x$ .
Hint 2
A coefficient is the number multiplying a power of $x$ . Use several distinct inputs, such as $1$ through $5$ : one input gives only one equation, which can't determine both $a$ and $b$ .
Hint 3
A regression fits unknown numbers to data. In Desmos, compare $ax_1^3+bx_1$ with the given expression at your list of inputs, then read $a$ and $b$ under PARAMETERS. Which subtraction does the question ask for?

Step-by-step

Fit the two coefficients in Desmos
Step 1
Turn equivalence into an identity
“Equivalent” means the expressions have the same value for every $x$ . So they form an identity, an equality that always holds: $(3x+2)^4-(3x-2)^4=ax^3+bx$ .
Step 2
Name the given expression
Type $f(x)=(3x+2)^4-(3x-2)^4$ in Desmos. A function gives an output for each input; Desmos saves the given expression as $f$ so you can use it in the fit.
Step 3
Choose distinct inputs
Type $x_1=[1...5]$ . Desmos makes the list $1,2,3,4,5$ . Use more than one input because a single equation can't determine both unknown coefficients.
Step 4
Find the cubic and linear coefficients
Type $ax_1^3+bx_1\sim f(x_1)$ . The tilde tells Desmos to run a regression, fitting $a$ and $b$ across the list. Under PARAMETERS, it shows $a=432$ and $b=192$ . These are the coefficients of $x^3$ and $x$ , respectively.
Step 5
Subtract in the requested order
Type $a-b$ . Desmos prints $240$ , the cubic coefficient minus the linear coefficient. Keep the stated order; $x=1$ would give $a+b$ instead. Grid in 240.

Question 57·SAT Desmos Regression Practice·Advanced Math

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