Question

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.Zeros of -2, 1, 0 and P(2) = 40

Answer 1

We are to find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, 0 and P(2) = 40

`\begin{gathered} x\text{ = -2} \\ x+2\text{ = 0} \end{gathered}`
`\begin{gathered} x\text{ = 1} \\ x\text{ - 1 = 0} \end{gathered}`
`x\text{ = 0}`

Let "a" be the coefficient be;

`\begin{gathered} P\text{ (x) = a (x + 2) (x - }1)\text{ x} \\ P(x)=a(x^3+x^2\text{ - 2x)} \\ P(x)\text{ = }ax^3+ax^2\text{ - 2ax} \end{gathered}`

P (2) = 40

`\begin{gathered} P(x)\text{ = a (x + 2) (x - }1)\text{ x} \\ P(2)\text{ = a (2 + 2) (2 -1) (2) = 40} \\ a\text{ (4)(1)(2) = 40} \\ 8a\text{ = 40} \\ (8a)/(8)=(40)/(8) \\ \\ a\text{ = 5} \end{gathered}`

`\begin{gathered} \\ P(x)\text{ = 5}x^3+5x^2\text{ - 2(5)x} \\ P(x)\text{ = 5}x^3+5x^2\text{ - 10x} \end{gathered}`