+0

# Find the zeros, HELP!!

0
11
5
+11

Find the sum and the product of the roots (real and complex) of x^3 + 3x^2 + 7x − 11 = 0

Mar 19, 2024

#5
+129829
+1

x^3 + 3x^2  + 7x  - 11   =  0

By Vieta

Sum of the roots  = -3 / 1  =   -3

Product of the roots = -(-11) /  1 =  11

Mar 20, 2024

#1
+60
+2

$$x^3 + 3x^2 + 7x − 11 = 0$$

simplifies to:

$$(x-1)(x^2+4x+11)=0$$

and

$$x^2+4x+11=0$$

which gives an imaginary number

but for the complex part,

you get the roots of x as:

$$x=\frac{-4±\sqrt{4^2-4*1*11}}{2}$$

whic becomes

$$x=2±\sqrt{-7}$$

so the roots of

$$x^3 + 3x^2 + 7x − 11 = 0$$

are

$$x = 1$$

and

$$x=2±\sqrt{-7}$$

Mar 19, 2024
#2
+11
0

Was there a specific method you used to factor x - 1 out? I understand (x-1)(x^2 + 4x + 11) = x^3 + 3x^2 + 7x − 11 but just looking at  x^3 + 3x^2 + 7x − 11, I do not know how to get the x-1

theadfas  Mar 19, 2024
edited by theadfas  Mar 19, 2024
#3
+60
+2

try it

hasAquestion  Mar 19, 2024
#4
+396
+1

In response to your question:

A good way to do this is by the rational root theorem.

For any polynomial, $$a_nx^n+a_{n-1}x^{n-1}\dots a_1x^1+a_0$$, the rational roots can all be expressed as:$$\pm\frac{\text{factor of }a_0}{\text{factor of }a_n}$$.

Therefore in this polynomial, the only possible rational roots are $$\pm 1, \pm11$$. (factors of a0 are 1, 11 and the only factor of ais 1)

Plug these values into the function, to see which ones produce zero. $$\pm1$$ is a really, really common ones, so always try those first.

Remainder theorem is always helpful. Remember if for a function $$f(x)$$$$f(a)=0$$, then $$x-a$$ is a factor of $$f(x)$$.

Here is the workthrough of the first step:

Use rational root theorem, plug in 1 to $$f(x)={x}^{3}+3x^{2}+7x-11$$.

$${1}^{3}+3*{1}^{2}+7*1-11=0$$.

Therefore we know because $$f(1) = 0$$, then x-1 is a factor of $$f(x)$$, so we can factor it out.

Mar 20, 2024
edited by hairyberry  Mar 20, 2024
#5
+129829
+1

x^3 + 3x^2  + 7x  - 11   =  0

By Vieta

Sum of the roots  = -3 / 1  =   -3

Product of the roots = -(-11) /  1 =  11

CPhill Mar 20, 2024