Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. x^3+2 x^2-5

Use the Rational Zero Theorem as an aid in finding all real...

Key Concepts

Synthetic Division

Synthetic Division is a streamlined method for dividing a polynomial by a linear factor of the form (x - c). It provides a quick way to test potential zeros (found using the Rational Zero Theorem) and to simplify the polynomial by factoring it, ultimately aiding in the complete determination of all zeros.

Polynomial Factorization

Polynomial factorization involves rewriting a polynomial as a product of simpler polynomials or linear factors. When a zero is identified, the polynomial can be expressed in terms of its factors, reducing its degree and simplifying the process of finding additional zeros.

Rational Zero Theorem

The Rational Zero Theorem states that for a polynomial with integer coefficients, any potential rational zero can be written in the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem narrows down the candidates for rational zeros, making it easier to test which ones are actual zeros of the polynomial.

Polynomial Zeros

Polynomial zeros are the solutions to the equation formed when setting a polynomial equal to zero. Finding these zeros is fundamental to understanding the function's behavior, as each zero represents an x-intercept on the graph, and knowing them helps in solving the polynomial equation completely.

Transcript

00:01 So we want to find the real solutions to the polynomial x -cube plus 2x squared minus 5x minus 6 using the rational zero theorem, which tells us if we consider these two letters p and q, where p is the factors of the constant, the constant term.

00:28 And then q are the factors of of the leading term.

00:36 So the leading coefficient.

00:40 Go ahead and write that better.

00:42 In this case, p would be the factors of negative 6 and q would be the factors of this implied 1.

00:51 So, and what this theorem tells us, if we take these information, we know that all rational zeros, so all rational zeros must be of the form, p divided a q.

01:11 Now, this doesn't tell us that we have have rational zeros per se, but it tells us that if there were to exist a rational zero, it must be of the form, a p divided by q.

01:21 And so in this case, we know p is just the factors of negative six.

01:26 So we have plus and minus one, plus and minus two, plus and minus three, and plus and minus six, and q is just the factors of one, which is plus and minus one.

01:36 Since we're considering only p divided by q, and q is only plus or minus one, all the p, values make up the all of the the p values that themselves make up all the possible rational zeros.

01:51 So what we can go ahead and do is since we don't have a really way to factor this, we can go ahead and use synthetic division plugging in each one by one using gas and check to see if we can find a solution to our equation, a zero, and the video of synthetic division is that if we do that and find a zero, we'll automatically find its factorization as well.

02:13 So let's go ahead and write this so we know we have x cubed plus 2x squared minus 5x minus 6.

02:24 And if we go ahead and start with x equals 1, we can go ahead and do synthetic division...

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