Home » Blog » Find a degree 2 polynomial to satisfy an integral equation

Find a degree 2 polynomial to satisfy an integral equation

Let P(x) be a quadratic polynomial such that

    \[ P(0) = P(1) = 0 \qquad \text{and} \qquad \int_0^1 P(x) \, dx  = 1. \]


Since P(x) is a quadratic polynomial we can write,

    \[ P(x) = ax^2 + bx + c. \]

Then, P(0) = 0 gives us c=0 and P(1) = 0 gives us

    \[ a + b + c = 0 \quad \implies \quad a = -b. \]

Finally, evaluating the integral we have

    \begin{align*}  \int_0^1 P(x) \, dx = 1 && \implies && \int_0^1 (ax^2 - ax) \, dx &= 1 \\ &&\implies && a \left. \left( \frac{x^3}{3} - \frac{x^2}{2} \right) \right|_0^1 &= 1 \\ &&\implies && -\frac{a}{6} &= 1 \\ &&\implies && a &= -6. \end{align*}

Thus, we have

    \[ P(x) = 6x- 6x^2. \]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):