Answer:
The solutions and the x-intercepts of the polynomial
are:
![x=\sqrt{10},\:x=-\sqrt{10},\:x=\sqrt{2},\:x=-\sqrt{2}](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B10%7D%2C%5C%3Ax%3D-%5Csqrt%7B10%7D%2C%5C%3Ax%3D%5Csqrt%7B2%7D%2C%5C%3Ax%3D-%5Csqrt%7B2%7D)
Step-by-step explanation:
Given a function <em>f</em> a solution or a root of <em>f</em> is a value
at which
.
An x-intercept is a point on the graph where y is zero.
To find the solutions of the polynomial and the x-intercepts
you need to:
First, we need to factor the polynomial expression
Factor the common term
![{\left(2 x^{4} - 24 x^{2} + 40\right)} = {\left(2 \left(x^{4} - 12 x^{2} + 20\right)\right)}](https://tex.z-dn.net/?f=%7B%5Cleft%282%20x%5E%7B4%7D%20-%2024%20x%5E%7B2%7D%20%2B%2040%5Cright%29%7D%20%3D%20%7B%5Cleft%282%20%5Cleft%28x%5E%7B4%7D%20-%2012%20x%5E%7B2%7D%20%2B%2020%5Cright%29%5Cright%29%7D)
We can treat
as a quadratic function with respect to ![x^2](https://tex.z-dn.net/?f=x%5E2)
Let
. We can rewrite
in terms of
as follows:
We need to solve the quadratic equation
![u^2-12u+20=0](https://tex.z-dn.net/?f=u%5E2-12u%2B20%3D0)
for this we can use the Quadratic Equation Formula:
For a quadratic equation of the form
the solutions are
![x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
![\mathrm{For\:}\quad a=1,\:b=-12,\:c=20:\quad u_{1,\:2}=\frac{-\left(-12\right)\pm \sqrt{\left(-12\right)^2-4\cdot \:1\cdot \:20}}{2\cdot \:1}](https://tex.z-dn.net/?f=%5Cmathrm%7BFor%5C%3A%7D%5Cquad%20a%3D1%2C%5C%3Ab%3D-12%2C%5C%3Ac%3D20%3A%5Cquad%20u_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-%5Cleft%28-12%5Cright%29%5Cpm%20%5Csqrt%7B%5Cleft%28-12%5Cright%29%5E2-4%5Ccdot%20%5C%3A1%5Ccdot%20%5C%3A20%7D%7D%7B2%5Ccdot%20%5C%3A1%7D)
![u_1=\frac{-\left(-12\right)+\sqrt{\left(-12\right)^2-4\cdot \:1\cdot \:20}}{2\cdot \:1}\\u_1=10](https://tex.z-dn.net/?f=u_1%3D%5Cfrac%7B-%5Cleft%28-12%5Cright%29%2B%5Csqrt%7B%5Cleft%28-12%5Cright%29%5E2-4%5Ccdot%20%5C%3A1%5Ccdot%20%5C%3A20%7D%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5Cu_1%3D10)
![u_2=\frac{-\left(-12\right)-\sqrt{\left(-12\right)^2-4\cdot \:1\cdot \:20}}{2\cdot \:1}\\u_2=2](https://tex.z-dn.net/?f=u_2%3D%5Cfrac%7B-%5Cleft%28-12%5Cright%29-%5Csqrt%7B%5Cleft%28-12%5Cright%29%5E2-4%5Ccdot%20%5C%3A1%5Ccdot%20%5C%3A20%7D%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5Cu_2%3D2)
the solutions to the quadratic equation are:
![u=10,\:u=2](https://tex.z-dn.net/?f=u%3D10%2C%5C%3Au%3D2)
Therefore, ![u^2-12u+20=(u-10)(u-2)](https://tex.z-dn.net/?f=u%5E2-12u%2B20%3D%28u-10%29%28u-2%29)
Recall that
so
![2 x^{4} - 24 x^{2} + 40=2 \left(x^{2} - 10\right) \left(x^{2} - 2\right)=0](https://tex.z-dn.net/?f=2%20x%5E%7B4%7D%20-%2024%20x%5E%7B2%7D%20%2B%2040%3D2%20%5Cleft%28x%5E%7B2%7D%20-%2010%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%202%5Cright%29%3D0)
Using the Zero factor Theorem: If ab = 0, then either a = 0 or b = 0, or both a and b are 0.
roots are
; ![x_2=-\sqrt{10}](https://tex.z-dn.net/?f=x_2%3D-%5Csqrt%7B10%7D)
roots are
; ![x_2=-\sqrt{2}](https://tex.z-dn.net/?f=x_2%3D-%5Csqrt%7B2%7D)
The solutions and the x-intercepts are:
![x=\sqrt{10},\:x=-\sqrt{10},\:x=\sqrt{2},\:x=-\sqrt{2}](https://tex.z-dn.net/?f=x%3D%5Csqrt%7B10%7D%2C%5C%3Ax%3D-%5Csqrt%7B10%7D%2C%5C%3Ax%3D%5Csqrt%7B2%7D%2C%5C%3Ax%3D-%5Csqrt%7B2%7D)
Because all roots are real roots the x-intercepts and the solutions are equal.