Question 99·Hard·Advanced Math

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Suppose

\frac{x^3 + ax^2 + bx + c}{x^2 + 3x + 2} = x + 4 + \frac{5}{x^2 + 3x + 2}

for all real values of $x$ such that the denominator is nonzero, where $a$ , $b$ , and $c$ are constants. What is the value of $b$ ?

Your answer

An equation true for all allowed $x$ becomes a polynomial identity when you clear its shared denominator. Coefficients of the same power of $x$ must match. You can also use a Desmos regression at several allowed inputs to find the missing constants. One input cannot determine three constants, and the leftover $5$ affects the constant term, not the $x$ term.

Hints

Hint 1
An identity holds at every allowed input, so you aren't solving for $x$ . The unknowns are $a$ , $b$ , and $c$ . Why would testing the equation at only one value of $x$ leave too much unknown?
Hint 2
A regression fits unknown constants across several inputs. In Desmos, use $x_1$ for a list of positive inputs and $\sim$ in place of $=$ . Positive inputs keep the denominator nonzero.

Step-by-step

Approach 1: Fit the constants with Desmos

Step 1
Choose allowed inputs
Type $x_1=[1...5]$ . Desmos lists $1,2,3,4,5$ . Here $x_1$ is your list of test inputs. Each is positive, so $x^2+3x+2$ is positive at each one; none makes the denominator zero.
Step 2
Fit the identity and read $b$
Type the given equation with every $x$ changed to $x_1$ and $=$ changed to $\sim$ . The regression asks Desmos to find constants that fit all five inputs. Under PARAMETERS, Desmos shows $a=7$ , $b=14$ , and $c=13$ . The question asks for $b$ , the coefficient of $x$ , so its value is $14$ . Grid in 14.

Approach 2: Match the $x$ terms by hand

Step 1
Clear the shared denominator
Multiply both sides by the nonzero denominator $x^2+3x+2$ . The two fractions lose that denominator, giving: $x^3+ax^2+bx+c=(x+4)(x^2+3x+2)+5$ .
Step 2
Find only the products containing $x$
Multiply $x$ by $2$ to get one $x$ term: $x(2)=2x$ . Multiply $4$ by $3x$ to get the other: $4(3x)=12x$ . The extra $5$ is a constant, so it adds no $x$ term.
Step 3
Match the coefficients
A coefficient is the number multiplying a power of $x$ . Because the equation holds for every allowed $x$ , the coefficients of $x$ must match. Set $b$ equal to the two contributions: $b=2+12$ . Add: $b=14$ . The value of $b$ is $14$ . Grid in 14.

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Question 99·Hard·Advanced Math

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