Question 106·Hard·Advanced Math

A cubic polynomial has highest power $3$, so write its rule as:

The four coefficients $a$, $b$, $c$, and $d$ are unknown; the table gives four input-output pairs to determine them.

Enter the table in Desmos, then type $y_1\sim ax_1^3+bx_1^2+cx_1+d$. The $\sim$ tells Desmos to fit the same coefficients to every row. Under PARAMETERS, it shows $a\approx-1.333333$, $b=4$, $c\approx1.333333$, and $d=-4$.

Type $P(x)=ax^3+bx^2+cx+d$, then $P(4)$. Desmos uses the fitted coefficients and prints $-20$, so the requested value is $P(4)=-20$. Grid in -20.

No Desmos needed. Equally spaced inputs let you use constant third differences. Each $x$ rises by $1$, so subtract neighboring outputs to get the first differences, or changes between values:

These changes are not constant, so don't keep adding $4$.

Subtract neighboring first differences to get the second differences:

The second differences change too.

Subtract the second differences to get the third difference:

For a cubic with equally spaced inputs, third differences stay constant. So the next third difference is also $-8$.

Add that repeated third difference to the last second difference:

Use the result to extend the last first difference:

Add this next change to the table's $P(3)=0$:

Grid in -20.