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F(x)=x^7-x^4 identify the zeros

User Issa Qandil
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1 Answer

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15 votes

ANSWER

x = 0 and x = 1

Step-by-step explanation

In the given function


f(x)=x^7-x^4

We can see clearly that one of the zeros is x = 0. Note that x is in all the terms of the polynomial, so whenever x is zero, the value of the function is zero.

According to the degree of this polynomial we should find 7 zeros. However, as we can see there are only two terms in this function and they all have x. Therefore, it is likely that the some of the zeros have a multiplicity greater than 1.

For x = 0, if we take x⁴ as a common factor:


f(x)=x^4(x^3-1)

We can see that the multiplicity of this zero is 4.

Another zero is found by solving:


x^3-1=0

Add 1 to both sides of the equation:


\begin{gathered} x^3-1+1=0+1 \\ x^3=1 \end{gathered}

And take cubic root:


\begin{gathered} \sqrt[3]{x^3}=\sqrt[3]{1} \\ x=1 \end{gathered}

The other zero is x = 1, with multiplicity 3.

Since 4+3=7, we have found all the zeros of this function.

User Jenna Kwon
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