Question #4a1e9

2 Answers
Nov 6, 2017

y=-1/272(x+5)(x^2+4x+13)

Explanation:

All cubic equations can be written in the form y=a(x-r_1)(x-r_2)(x-r_3), where a is a constant and r_1,r_2,r_3 are roots of the equation.

Since we know the roots, we can write the cubic equation as y=a(x+5)(x+2+3i)(x+2-3i) (the complex roots of all real polynomials occur in conjugate pairs; thus since one of the roots is -2-3i, another root is -2+3i).

Simplifying this gives y=a(x+5)(x^2+4x+13).

It is given that y(3)=-1, then -1=a(3+5)((3)^2+4*3+13)=272a, or a=-1/272.

The final answer is y=-1/272(x+5)(x^2+4x+13).

Nov 6, 2017

Equation for f(x):
f(x) = -1/112(x+5)(x^2+5)

Explanation:

Because one of the zeros is irrational in a way that it includes i, there must be another zero that includes the conjugate of that term.

f(x) = a(x+5)(x-(-3i-2))(x+(-3i-2))

f(x) = a(x+5)(x+3i+2)(x-3i-2)

f(x) = a(x+5)(x^2 cancel(-3ix)cancel(-2x) cancel(+3ix)+9 cancel(-6i) cancel(+2x) cancel(-6i) -4)

f(x) = a(x+5)(x^2+5)

Now use the given to find a: f(3)=-1
-1 = a(3+5)(3^2+5)

-1=8*14*a

a=-1/(112)

Equation for f(x):
f(x) = -1/112(x+5)(x^2+5)