Two zeros of the polynomial p(x)=x4−4x3−24x2+20x+7 are 1 and 7. The other two zeros are real, but irrational. Divide p by two linear factors or by one quadratic factor.

Answer 1

divide by x^2 - 8x +7

or by (x-1) and then again by (x-7)

divide by the linear factors and you can use synthetic division

1 1 -4 -24 20 7

1 -3 -27 -7 0 means x^3- 3x^2 -27x -7 = quotient

then

7 1 -3 -27 - 7

1 4 1 0 means x^2 +4x +1 is the final quotient

it doesn't factor, easily, so complete the square

x^2 +4x +4 = -1 +4

(x+2)^2 = 3

x+2 = + or - sqr3

x = -2 +/-sqr3 are the two irrational roots

= about -3.732 or .-268

you could also graph the quadratic quotient or the original 4 degree polynomial

and find the x intercepts = roots = 1, 7, -2+sqr3, -2-sqr3

Answer 2

Hey LPL,

have you divided the long equation by the quadratic of (x-1)(x-7), which is (x²-8x+7)?

as the instructions say, you can do that or divide by (x-1) and then divide the quotient by (x-7).

Try that and report your results, and we can help you further.

if you don’t know how to divide by a binomial, then please ask here.