Question 258·Medium·Advanced Math

The words can be written in the form mean the two expressions are an identity: they agree for every $x$. A coefficient is a number multiplying a power of $x$. Type $x_1=[1...5]$; Desmos lists $1,2,3,4,5$. These distinct inputs let the next line fit the three unknown coefficients.

Type $(3x_1-2)(ax_1^2+bx_1+c)\sim15x_1^3+14x_1^2-7x_1-6$. The $\sim$ asks Desmos to fit the letters using your input list. Under PARAMETERS, Desmos reports $a=5$, $b=8$, and $c=3$. The requested coefficient is $b$, so its value is $8$, not the $14$ attached to $x^2$ in the expanded polynomial. Grid in 8.

Only $(3x)(ax^2)$ produces an $x^3$ term in the product. Its coefficient, the number multiplying $x^3$, must match the given $x^3$ coefficient:

Divide by $3$:

Two products make $x^2$ terms: $(3x)(bx)=3bx^2$ and $(-2)(ax^2)=-2ax^2$. Keep the minus sign from $-2$. Their combined coefficient must match the given $14$:

Use $a=5$ to find the coefficient the question asks for. Substitute:

Multiply:

Add $10$:

Divide by $3$: $b=8$. The value of $b$ is $8$. Grid in 8.