In Exercises 39–40, graph each polynomial function. f(x)=-x^3(x+4)^2(x-1)

Name: Problem 40, Chapter 3: Precalculus
Uploaded: 2020-06-19T11:56:31-08:00
Duration: 7 min 18 s
Channel: Ankit Gupta
Description: In Exercises 39–40, graph each polynomial function. f(x)=-x^3(x+4)^2(x-1)

In Exercises 39–40, graph each polynomial function....

Key Concepts

Polynomial Functions

A polynomial function is an expression constructed from variables and coefficients using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Such functions are continuous everywhere and exhibit smooth, unbroken curves as their graphs, which makes them predictable in terms of their behavior between and at their roots.

Zeros and Multiplicities

The zeros (or roots) of a polynomial are the values of the variable at which the function equals zero, corresponding to the points where the graph intersects or touches the x-axis. The multiplicity of each zero indicates how many times a particular factor occurs; an odd multiplicity means the graph crosses the axis at that zero, while an even multiplicity means the graph merely touches and turns around at that point.

Factoring Polynomials

Factoring is the process of rewriting a polynomial as a product of its simpler polynomial factors. This is a crucial strategy for finding the zeros of the function and for understanding its structure, as it readily reveals both the roots and their respective multiplicities, facilitating the overall graphing process.

End Behavior of Polynomials

The end behavior of a polynomial function describes how its values behave as the input grows larger in the positive or negative direction. This behavior is determined by the leading term (the term with the highest power), which dictates whether the graph will rise or fall at the extremes of the x-axis.

Transcript

00:01 A polynomial function f of x is given as x to the power 4 plus x to the power 3 minus 3 x squared minus x plus 2 so for this polynomial function we have to plot its graph so for plotting the graph first we have to find its real zeros so for finding real zeros first we have to find out how many number of possible rational zeros so for rational zeros we will use rational zeros we will use rational zeros theorem which state that the rational zeros will be in the form of p divided by q where p are the factors of constant of given polynomial so the value of p will be equal to the factors of two so the factors of two will be equal to plus minus one and plus minus two like this the value of q are the factors of the leading coefficient here leading coefficient is one so the value of q will be equal to plus minus one from here the value of p divide by q equal to plus minus 1 and plus minus 2 and these are the possible additional zeros out of them we have to find the real zeros so if we put the value of x equal to 1 in the polynomial we will get 1 plus 1 minus 3 minus 1 plus 2 so this will give 0 value now if we put the value of x as minus 1 we will get 1 minus 1 minus 1 minus 3 plus 1 plus 2 so from here we will get 0 so by these two questions we will say x minus 1 will be factor of this polynomial and x plus 1 will be also a factor of given polynomial now we have find out two more factor of given polynomial because this is 4 degree polynomial so here are the at most 4 so for finding two more zeros, we will divide the polynomial by the product of these two factors.

02:11 So here we will get after dividing x to the power 4 plus x to the power 3 minus 3 x squared minus x plus 2.

02:27 So this will be multiplied by x square.

02:30 So this will after multiply we will get x to the power 4 minus x square.

02:35 So on substation sign will be changed and here will be xx q and this will be minus 2 x square this will be multiplied by plus x and here we will get x q this will be minus x on succession sign will be changed this will be minus 2 x square if we put the value of x as plus x and here will be minus x plus 2 so x will be cancel out to each other now this will be multiplied by minus 2...

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