Question

Find a polynomial of the specified degree that satisfies the given conditions. degree 4; zeros −5, 0, 1, 6; coefficient of x3 is 4

Answer 1

We have to determine the polynomial of degree 4 with zeros −5, 0, 1, 6 and the coefficient of $x^3$ is 4.

Since the zeros of the given polynomial are -5,0,1 and 6.

Therefore, $(x+5)(x-0)(x-1)(x-6) =0$

$(x^2+5x)(x-1)(x-6)=0$

$(x^3-x^2+5x^2-5x)(x-6)=0$

$(x^3+4x^2-5x)(x-6)=0$

$x^4+4x^3-5x^2-6x^3-24x^2+30x = 0$

$x^4-2x^3-29x^2+30x=0$

Since the coefficient of $x^3$ is 4, therefore, we get

$-2x^4+4x^3+58x^2-60x = 0$ is the required polynomial with degree 4, coefficient  of $x^3$ as 4 and zeros as -5,0,1 and 6.