Let P(x)=x^2+bx+c be a quadratic polynomial with real coefficients, such that _0^1P(x),dx=…

Question

Let $P(x)=x^2+bx+c$ be a quadratic polynomial with real coefficients, such that $\displaystyle\int_0^1P(x)\,dx=1$ and $P(x)$ leaves remainder 5 when divided by $(x-2)$. Then the value of $9(b+c)$ is

Accepted Answer

$\displaystyle\int_0^1(x^2+bx+c)\,dx=\frac13+\frac b2+c=1 \implies b+2c=\frac43.$

$P(2)=5 \implies 4+2b+c=5 \implies c=1-2b$.

Substituting: $b+2(1-2b)=\dfrac43 \implies -3b+2=\dfrac43 \implies b=\dfrac29$, so $c=\dfrac59$.

$$9(b+c)=9\left(\frac29+\frac59\right)=7.$$

Solutions · 1