Question

Find a degree 4 polynomial having zeros -7.-4.3 and 5 and the coefficient of x^4 equal 1

The polynomial is​

Answer 1

`P(x)=(x+7(x+4)(x-3)(x-5)=x^(4)+3x^(3)-45x^(2)-59x+420`

Explanation:

Given:

The zeros of the polynomial are -7, -4, 3 and 5.

The coefficient of
`x^(4)` is 1.

A polynomial of degree 4 has maximum 4 zeros.

Let the polynomial be
`P(x)`.

Since,
`P(x)` has zeros -7, -4, 3 and 5, therefore,

`P(x)=a(x-(-7))(x-(-4))(x-3)(x-5)\\P(x)=a(x+7)(x+4)(x-3)(x-5)`

Where,
`a` is coefficient of
`x^(4)`.

But, coefficient of
`x^(4)` is 1. So,
`a=1`

∴
`P(x)=(x+7)(x+4)(x-3)(x-5)\\P(x)=(x^(2)+11x+28)(x^(2)-8x+15)\\P(x)=x^(4)-8x^(3)+15x^(2)+11x^(3)-88x^(2)+165x+28x^(2)-224x+420\\P(x)=x^(4)+3x^(3)-45x^(2)-59x+420`