Question

Find a polynomial of the specified degree that satisfies the given conditions. Degree 4; zeros −1, 1, 6 ; integer coefficients and constant term 18 f(x) =

Answer 1

Answer:

${x}^{4} - 3 {x}^{3} - 19 {x}^{2} + 3x + 18$

Step-by-step explanation:

If a is a zero of a polynomial then

$(x - a)$

is a factor of the polynomial.

So since, -1, 1 ,6 are thr zeroes, then the factors is

$(x + 1)(x - 1)(x - 6)$

$( {x}^{2} - 1)(x - 6)$

${x}^{3} - 6 {x}^{2} - x + 6$

We aren't done because we have a degree of 3 and a constant of 6.

Since we want a constant term 18, and a additional degree, we multiply by another binomial.

which will be

(x+3),

$({x}^{3} - 6 {x}^{2} - x + 6)(x + 3)$

${x}^{4 } - 6 {x}^{3} - {x}^{2} + 6x + 3 {x}^{3} - 18 {x}^{2} - 3x + 18$

Which simplified gives us

${x}^{4} -3 {x}^{3} - 19 {x}^{2} + 3x + 18$