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Find a polynomial $p$ of degree $3$ such that

\begin{align*} p(−4) &= −142, \\ p(1) &= −2, \\ p(−5) &= −242, \\ p(4) &= 10. \end{align*}

Then use your polynomial to approximate $p(2)$.

\begin{align*} p(x) &= ?, \\ p(2) &= ?. \end{align*}

I can't find this sort of question in the textbook so I'm having trouble. Please teach me how to solve this question and, perhaps, fill in point 2?

Thank you.

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2 Answers

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You general cubic has the form $p(x) = ax^3 + bx^2 + cx + d$.

Plugging in the four points will give you four equations in the four unknowns $\{a,b,c,d\}$.

For example, $p(1) = -2 \Rightarrow -2 = a + b + c + d$.

Solve that system of equations however you know best!

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Hint: Find constants $a$, $b$, $c$ and $d$ such that the polynomial $$a(x-1)(x+5)(x-4)+b(x+4)(x+5)(x-4)+c(x+4)(x-1)(x-4)+d(x+4)(x-1)(x-5)$$ takes on the desired values.

For example, to compute $a$, we want $a(-4-1)(-4+5)(-4-4)=-142$.

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