How do you reduce a third degree term to fit quadratic formula?

Question

I have the first root but to use quadratic formula to simplify the second and third root from the other factor I would have to get the bottom right term to a degree of 2 or less without compromising the (a)(x)^2+(b)(x)+c form needed to substitute into the quadratic formula. Or do I have the problem wrong all together and I need to use cube method first or something? the end goal is to find all 3 roots but I only need help getting rid of that pesky third degree, I know how to do the rest if it would only substitute.

The equation: x^3+x^2+4x+4

The first zero after synthetic dividing: (-1,0)

The term in question that I derived from the synthetic division: x^3+4x

Just help me reduce it so it can fit quadratic formula.

Answer 1

Hi Paul,

Concerning your problem, another way of doing it is to find one root by trial and error. In this case you’ll have x = -1 as one root.

Then you do long division in other to obtain the other two roots.

After dividing, you’ll obtain (x+1)(x^2 +4). But (x^2 +4) has no real solution.

Therefore there is only one real solution to your problem, which is (x=1). The other two solutions are complex.

Answer 2

x^3 + x^2 + 4x + 4 = 0

x^2(x+1) + 4(x+1) = 0

(x+1)(x^2 + 4) = 0

x+1 = 0 or x^2 + 4 = 0

x=-1 or x^2 = -4

x = +or- 2i

You can also use synthetic division:

-1 | 1 1 4 4

-1 0 -4

________________________________

1 0 4 0

remainder is 0

constant term is 4

linear coefficient is 0

quaratic coefficient is 1

so the resulting quadratic is x^2 + 4

which has solutions x = +or- 2i as show above

It can also be factored over the imaginaries:

x^2 + 4 = x^2 - (-4)

= ( x + 2i)(x - 2i) <--difference of squares pattern

checks the solutions:

x=-1: (-1)^3 + (-1)^2 + 4(-1) + 4 =

-1 + 1 + -4 + 4 = 0

x= 2i:

(2i)^3 + (2i)^2 + 4(2i) + 4 =

8(-i) + 4(-1) + 8i + 4 =

=8i + -4 + 8i + 4 = 0

x= -2i will also check per conjugate theorem..

but here it is anyway: (-2i)^3 + (-2i)^2 + 4(-2i) + 4 =

(-8)(-i) + (4)(-1) + 4(-2i) + 4 =

8i + -4 + -8i + 4 = 0