Question 113794

{{{f(x)=x^4-4x^2}}} Start with the given function



{{{f(x)=x^2(x^2-4)}}} Factor out the GCF {{{x^2}}}



{{{f(x)=x^2(x-2)(x+2)}}} Factor {{{x^2-4}}} using the difference of squares to get {{{(x-2)(x+2)}}}



Now to find the x-intercepts, set {{{f(x)}}} equal to zero and solve for x



{{{x^2(x-2)(x+2)=0}}} 


Break up the equation using the zero product property

{{{x^2=0}}} , {{{x-2=0}}} or {{{x+2=0}}}



{{{x=0}}} , {{{x=2}}} or {{{x=-2}}} Solve for x in each case



So our solutions are {{{x=0}}} , {{{x=2}}} or {{{x=-2}}} which means the x-intercepts are (0,0), (2,0), and (-2,0)



Notice if we graph, we get


{{{ graph( 500, 500, -10, 10, -10, 10, x^4-4x^2) }}}


and we can see that the x-intercepts are (0,0), (2,0), and (-2,0)



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Let's evaluate {{{f(-3)}}}

{{{f(x)=x^4-4x^2}}} Start with the given function.



{{{f(-3)=(-3)^4-4(-3)^2}}} Plug in {{{x=-3}}}. In other words, replace each x with -3.



{{{f(-3)=(81)-4(-3)^2}}} Evaluate {{{(-3)^4}}} to get 81.



{{{f(-3)=(81)-4(9)}}} Evaluate {{{(-3)^2}}} to get 9.



{{{f(-3)=(81)-36}}} Multiply -4 and 9 to get  -36



{{{f(-3)=45}}} Now combine like terms



-----------Now let's evaluate another value---------

Let's evaluate {{{f(-1)}}}

{{{f(x)=x^4-4x^2}}} Start with the given function.



{{{f(-1)=(-1)^4-4(-1)^2}}} Plug in {{{x=-1}}}. In other words, replace each x with -1.



{{{f(-1)=(1)-4(-1)^2}}} Evaluate {{{(-1)^4}}} to get 1.



{{{f(-1)=(1)-4(1)}}} Evaluate {{{(-1)^2}}} to get 1.



{{{f(-1)=(1)-4}}} Multiply -4 and 1 to get  -4



{{{f(-1)=-3}}} Now combine like terms



-----------Now let's evaluate another value---------

Let's evaluate {{{f(1)}}}

{{{f(x)=x^4-4x^2}}} Start with the given function.



{{{f(1)=(1)^4-4(1)^2}}} Plug in {{{x=1}}}. In other words, replace each x with 1.



{{{f(1)=(1)-4(1)^2}}} Evaluate {{{(1)^4}}} to get 1.



{{{f(1)=(1)-4(1)}}} Evaluate {{{(1)^2}}} to get 1.



{{{f(1)=(1)-4}}} Multiply -4 and 1 to get  -4



{{{f(1)=-3}}} Now combine like terms



-----------Now let's evaluate another value---------

Let's evaluate {{{f(3)}}}

{{{f(x)=x^4-4x^2}}} Start with the given function.



{{{f(3)=(3)^4-4(3)^2}}} Plug in {{{x=3}}}. In other words, replace each x with 3.



{{{f(3)=(81)-4(3)^2}}} Evaluate {{{(3)^4}}} to get 81.



{{{f(3)=(81)-4(9)}}} Evaluate {{{(3)^2}}} to get 9.



{{{f(3)=(81)-36}}} Multiply -4 and 9 to get  -36



{{{f(3)=45}}} Now combine like terms