Roots (Zeros)

a. Plot the ze...

Question

Roots (Zeros)

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²

Accepted Answer

Welcome back everyone. For the function G of X equals x to the power of 4 minus 4 x cubed plus 4 x squad, plot the zeros of G of X and the zeros of its derivative G of X. Let's begin with the first part of the problem where we want to identify the zeros of G of X. This means that we're setting G of X equals 0. We have X to the power of 4 minus 4 x cubed plus 4 x 2 this equals 0. Now, what we want to do is perform a basic factorization, and we can take out X2, which leaves us with X2 minus 4 X plus 4 MCs. We can factor out further because we have a perfect square, right? X2 minus 4 x + 4 gives us X minus 22. Setting it equal to 0, we have to understand that either our first factor is equal to 0, which gives us a solution of X equals 0, or our second factor of 0. So X minus 2 can also be equal to 0, right, which gives us our second solution of X equals 2, because 2 minus 2 gives us 0. So we're going to add those two points on the number line. We have X of 0 and X of 2. Well then, we have solved the first part, and now we can move on to the second part where we have to identify the derivative G of X and set it equal to 0. Let's begin by performing differentiation on X to the power of 4 minus 4 X cubed plus 4 X2. Let's differentiate using the power rule. We're going to get 4 X cubed minus 12 x 2 + 8X. And now we can perform factorization once again, we're going to take out 4 X, which leaves us with X2 minus 3X plus 2. We can factor out further because we have a quadratic polynomial, right? And this is going to give us for X multiplied by X minus 1, multiplied by X minus 2. Let's consider our factors x minus 1. If that's equal to 0, well, we're going to get X equals 1, or we can have our factor X minus 2 equals 0, which means that X equals 2. Additionally, we have our third factor for X, so to make G of X equals to 0, our X must be 0. So we have 3 additional 001 and 2. We already have 0 and 2 on the number line. So all that we have to do is simply add an additional X equals 1 value on the number line to finish this problem. Thank you for watching.

Roots (Zeros)

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²

Key Concepts

Roots (Zeros) of a Polynomial

First Derivative and Critical Points

Graphical Representation of Zeros

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