Find all of the real and imaginary zeros for each polynomial function. M(t)=18 t^3-21 t^2+10 t-2

Name: Problem 61, Chapter 3: Precalculus Functions and Graphs
Uploaded: 2020-04-08T00:47:05-08:00
Duration: 4 min 19 s
Channel: Bobby Barnes
Description: Find all of the real and imaginary zeros for each polynomial function. M(t)=18 t^3-21 t^2+10 t-2

Find all of the real and imaginary zeros for each polynomial...

Key Concepts

Rational Root Theorem

The Rational Root Theorem provides a method to list all possible rational roots of a polynomial with integer coefficients. By considering the factors of the constant term and the leading coefficient, one can generate a set of candidate roots to test, which is particularly useful when attempting to factor the polynomial and identify its zeros.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This guarantees that a polynomial of degree n will have exactly n roots in the complex number system, when counted with multiplicity, and underlies the search for both real and imaginary zeros.

Synthetic Division

Synthetic division is a simplified form of polynomial division that is used to divide a polynomial by a linear factor. This technique is particularly useful when testing candidate roots obtained via the Rational Root Theorem, as it efficiently determines whether a candidate is indeed a zero and helps reduce the polynomial to a lower degree for further analysis.

Polynomial Functions

Polynomial functions are expressions that involve sums and products of variables raised to whole number powers. They are fundamental objects in algebra and calculus, serving as a basis for understanding more complex functions. Analyzing these functions involves studying their behavior, including their roots, end behavior, and turning points.

Zeros of Polynomial Functions

The zeros of a polynomial function are the values of the variable that make the polynomial equal to zero. These zeros can be real or complex (imaginary) and they represent the x-intercepts of the graph of the function. Finding zeros is key in solving equations and understanding the factorization and graph behavior of the function.

Transcript

00:01 We want to find all of the real and imaginary zeros for our polynomial function here.

00:06 So just so we're not randomly guessing thousands of different numbers to hopefully get a solution, what we're going to do is use the rational zero theorem, which says, so this is going to be p, this is q, and now we will look at the factors of p over the factors of q so we'll have plus or minus over so it doesn't matter to that it is negative two so we'll just look at the factors of two because we have that plus or minus on the else side so we have one two and then for q is going to be 1, 18, 2, 9, 3, and 6.

00:58 And i believe that is all of the different factors.

01:05 Now, let's go ahead and decide what values from here we could use.

01:14 And i think it may be a good idea to use one half.

01:19 So normally we would go through the whole process of taking all these, dividing them into our numerator.

01:24 But that may take a while, so just for the sake of brevity, i'm going to make the guess that t is equal to one half, and it seems our divide inspiration will work out, at least we hope it does.

01:43 Now we go ahead and drop 18, multiply those together to get 9.

01:48 Adding that should give us so that is negative 12 multiplying one half by negative 12 gives us negative 6 adding that will give us 4 then multiplying by 1 half gives us 2 and we get 0 so it looks like our divine inspiration worked out now the reason why we wanted to synthetically divide it as opposed to just keep plugging values in because now we get this quadratic equation which should be one of our factors of m of t which means this will also have zeros of our polynomial so let's set that equal to zero now all of these are even so let's just go ahead and divide this by two or multiply it by one half which is going to give us 9x squared minus 6x plus 2 is equal to and now we can use the quadratic equation to help us find these zero.

02:54 So i'll go ahead and do that up here.

02:56 So it's going to be x is equal to negative b plus or minus b squared minus 4 a c, all of this square rooted all over 2a.

03:07 And remember, a, b, and c will be those coefficients for our quadratic...

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