The polynomial P(x)=x2+ax +b has a zero at x =2. When P(x)is divided by x +1, the remainder is 18.Find the values of a and b.

with working out please

You have \(\displaystyle P(x) = x^2 + ax + b\).

By the Remainder and Factor Theorems:

If there is a root at \(\displaystyle x = 2\), then \(\displaystyle P(2) = 0\).

If, when you divide by \(\displaystyle x + 1\) you get a remainder of \(\displaystyle 18\), then \(\displaystyle P(-1) = 18\).

So you have:

\(\displaystyle 2^2 + 2a + b = 0\)

\(\displaystyle (-1)^2 - a + b = 18\).

Simplify and solve these equations simultaneously for \(\displaystyle a\) and \(\displaystyle b\).