Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)^4(x-3), the number 2 is a zero of multiplicity 4.

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Hey, everyone here we are asked to identify whether the given statement is true or not. And if it is false, we are asked to write the reason. Here we are given five is a zero of the function F of X is equal to the quantity of X plus five raised to the fifth power times the quantity of X minus seven. And there is a multiplicity of five. Here we have two answer choice options, answer choice, a true and answer choice B false five is not a zero of the function. So our first step in this problem is to determine if five is in fact a zero of our function. So we just need to set our function F of X equal to zero. So doing this, we get the equation, the quantity of X plus five raised to the fifth power times the quantity of X minus seven is equal to zero. So now we just want to find the values of X that set the left side of our expression equal to zero. And so that means we just need to set each part of the expression individually equal to zero. So we can start with the first part of the expression where we have the quantity of X plus five raised to the fifth power. And again, we're setting this equal to zero and solving for X. So first, we need to get rid of this fifth power and we do this by taking the fifth root of both sides of the equation. And we know that the fifth route of zero just gives us zero. So we are left with X plus five is equal to zero. Now solving further four, X, we just need to subtract five from both sides to get X alone and we get X is equal to negative five. Now we just need to repeat this process with the other part of our original expression. So we now take X minus seven and set it equal to zero. Now solving for X, we just add seven to both sides and we get X is equal to seven. Well, here we see we have come up with X being equal to negative five or seven, but we do not have an X being equal to positive five. So we know since X is not equal to five, that our given statement is false since five can then not be a zero of our given function. So with this, we are left with answer choice B where our given statement is false because five is not a zero of our function. Thanks for watching. I hope you found this. Video helpful and I'll see you again next time.

Determine whether each statement is true or false. If false, explain why. For ƒ(x)=(x+2)^4(x-3), the number 2 is a zero of multiplicity 4.

Verified Solution

Key Concepts

Zeros of a Function

Multiplicity of Zeros

Polynomial Behavior at Zeros