Find all zeros of the polynomial function or solve the given polynomial equation. Use the Ration

Find all zeros of the polynomial function or solve the given...

Key Concepts

Polynomial Zeros

Polynomial zeros are the solutions to the equation f(x) = 0, where f(x) is a given polynomial. They represent the x-values where the graph of the polynomial crosses or touches the x-axis. The process of finding these zeros is fundamental in algebra and calculus, offering insight into the behavior of the function and its factors.

Rational Zero Theorem

The Rational Zero Theorem provides a method for identifying all possible rational zeros of a polynomial with integer coefficients by considering the factors of the constant term and the leading coefficient. This theorem streamlines the search for actual zeros by listing a finite set of candidates that can be tested, which is particularly useful when working with higher degree polynomials.

Descartes's Rule of Signs

Descartes's Rule of Signs is a technique to estimate the number of positive and negative real zeros of a polynomial based on the number of sign changes in the sequence of its coefficients. This rule gives an upper bound on the number of real zeros and helps in planning an effective strategy to isolate and test potential zeros.

Graphical Analysis of Polynomial Functions

Graphical analysis involves using graphing tools or sketching the function to visualize the behavior of a polynomial. By observing the graph, one can locate approximate positions of the zeros, examine the shape of the curve, and identify patterns that support analytical methods. This approach complements algebraic techniques, especially in complex equations.

Polynomial Factorization and Synthetic Division

Once potential zeros are identified, polynomial factorization is used to break down the polynomial into simpler components. Synthetic division is a streamlined method of dividing polynomials by linear factors and is especially useful for verifying candidate zeros obtained from the Rational Zero Theorem, thereby simplifying the process of finding all zeros.

Transcript

00:01 So in this question, we're going to look for the zeros of the function x to the fourth minus 4x cubed minus x squared plus 14x plus 10.

00:09 And so let's hop in to using the rational zero test.

00:14 Okay.

00:16 So if i think about the rational zero test, i'm thinking about possible rational zeros.

00:24 And possible rational zeros are going to look like factors of the constant term.

00:31 So in this case, factors of 10 being divided by factors of the leading coefficient, in this case, my leading coefficient is 1.

00:43 Now, what are my factors of 10? my factors of 10 are plus or minus 1, plus or minus 2, plus or minus 5, and plus or minus 10, plus or minus 10, all over the factors of one, which are plus or minus one.

01:05 And just to make that a little bit clearer, this should be a plus or minus 10 here at the end.

01:12 And so, since i'm just dividing by plus or minus one, that's giving us possible rational zeros of plus or minus one, plus or minus two, plus or minus five, and plus or minus 10.

01:26 And so with that of mind, let's go ahead.

01:29 Let's go ahead and start testing these rational zeros using synthetic division.

01:34 So i have this quartic polynomial, and so i'm going to write down my coefficient 1, negative 4, negative 1, positive 14, and positive 10.

01:49 And i may as well start with 1 and see what happens.

01:54 So let's see.

01:55 If i start with one, i drop down one, i multiply by one.

02:01 I add, i multiply by one.

02:05 I add, i multiply by one.

02:09 I add, i multiply by one.

02:13 And so close, but 10 plus 10 does not give me zero.

02:18 That's not going to be helpful.

02:20 Okay, so one is not a zero this time.

02:24 Remember, my goal is at the end of the synthetic division, i should be ending up with zero.

02:32 So since one didn't work, let's try negative one.

02:36 That's the next possibility that is suggested by my rational zero test.

02:42 Drop down the one, multiply by negative one, add, multiply by negative one, add, multiply by negative one, add, multiply by negative one, add, and now i'm getting zero.

03:05 And so negative one will be a zero of this polynomial.

03:10 And so that means that one of my factors of this polynomial will in fact be the quantity of x plus one.

03:20 What will my other factor be? well, it'll be a cubic factor.

03:24 Since i started with a quartic polynomial, and i've pulled out a linear guy, my remaining factor is going to be x cubed...

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