Question 64·SAT Desmos Regression Practice·Advanced Math

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For $x \ne 3$ , the expression

\frac{(2x^2-18)^2-(2x^2+18)(x-3)^2}{x-3}

is equivalent to $ax^3+bx^2+cx+d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

What is the value of $b$ ?

Your answer

A polynomial identity gives the same value at every allowed input, even when one form starts as a fraction. If a long expression is promised to equal a cubic, a Desmos regression can find its coefficients from several distinct inputs. Leave out any input that makes the original denominator zero. Reading a coefficient from the numerator before dividing is the tempting slip.

Hints

Hint 1
The denominator makes $x=3$ off limits, even if a factor later cancels. To fit the polynomial in Desmos, choose distinct inputs that avoid $3$ .
Hint 2
A coefficient is the number multiplying a power of $x$ . The promised cubic has four coefficients, so use at least four distinct allowed inputs to pin them down.

Step-by-step

Approach 1: Fit the promised cubic in Desmos
Step 1
Name the original expression
Equivalent means equal at every input the original allows. Since $x=3$ makes the denominator zero, keep it out of the fit. Type $f(x)=\frac{(2x^2-18)^2-(2x^2+18)(x-3)^2}{x-3}$ in Desmos; the next steps will use $f$ to refer to the whole fraction.
Step 2
Choose allowed inputs
Type $x_1=[-2,-1,0,1,2]$ ; Desmos shows your list of five distinct inputs. None is $3$ . Four distinct inputs determine the four coefficients of a cubic, and the fifth gives the fit another value to match.
Step 3
Read the requested coefficient
Type $f(x_1)\sim ax_1^3+bx_1^2+cx_1+d$ . The $\sim$ tells Desmos to fit the coefficients to the listed outputs. Under PARAMETERS, it shows $b=18$ . Since $b$ multiplies $x^2$ , the requested value is $18$ . Grid in 18.

Approach 2: Match only the highest powers
Step 1
Find the first square's leading term
The first square contains only even powers of $x$ , so it has no $x^3$ term. Its highest term is $(2x^2)^2$ . Keep the powers that can affect the requested coefficient:
$(2x^2-18)^2=4x^4+\text{terms of degree at most 2}.$
Step 2
Square the shorter factor
To find the second product's $x^4$ and $x^3$ terms, first expand both copies of $x-3$ :
$(x-3)^2=x^2-6x+9.$
Step 3
Keep the second product's highest terms
Multiply $2x^2$ by $x^2$ and by $-6x$ . Products involving $18$ reach at most $x^2$ , so they can't change the two highest terms:
$(2x^2+18)(x^2-6x+9)=2x^4-12x^3+\text{terms of degree at most 2}.$
Step 4
Subtract the two highest terms
Subtract the whole second product, including its negative $x^3$ term:
$\text{numerator}=4x^4-(2x^4-12x^3)+\text{terms of degree at most 2}.$
Combine those terms:
$\text{numerator}=2x^4+12x^3+\text{terms of degree at most 2}.$
Step 5
Account for division by $x-3$
For $x\ne3$ , multiplying the promised cubic by $x-3$ gives the numerator. Distribute only far enough to reach $x^4$ and $x^3$ :
$(x-3)(ax^3+bx^2+cx+d)=ax^4+(b-3a)x^3+\text{terms of degree at most 2}.$
The divisor mixes adjacent coefficients: the numerator's $x^3$ coefficient is $b-3a$ , not $b$ .
Step 6
Match the $x^4$ coefficients
Both forms describe the same numerator, so their coefficients of $x^4$ must match. The earlier numerator has $2x^4$ :
$a=2.$
Step 7
Solve for the $x^2$ coefficient
Match the numerator's $12x^3$ term:
$b-3a=12.$
Replace $a$ with $2$ :
$b-3(2)=12.$
Multiply:
$b-6=12.$
Add $6$ :
$b=18.$
So the coefficient of $x^2$ is $18$ . Grid in 18.

Question 64·SAT Desmos Regression Practice·Advanced Math

Strategy

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Step-by-step

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