Use the Rational Zero Theorem as an aid in finding all real zeros of the polynomial. 3 x^3+4 x^2

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

Key Concepts

Polynomial Division (Synthetic Division)

Polynomial division, including synthetic division, is a method used to divide a polynomial by a binomial of the form (x - c). This technique is important for confirming whether a candidate zero (c) is indeed a root and for reducing the polynomial's degree, which simplifies the process of finding the remaining zeros.

Real Zeros of Polynomials

Real zeros (or roots) are the x-values at which the polynomial evaluates to zero. Identifying all real zeros is critical to understanding the behavior of the polynomial’s graph, including where it crosses or touches the x-axis. Methods such as applying the Rational Zero Theorem and polynomial division are typically used in this process.

Rational Zero Theorem

This theorem provides a list of possible rational zeros for a polynomial with integer coefficients by taking the factors of the constant term and the factors of the leading coefficient. It helps narrow down the candidates that can be tested as actual zeros of the polynomial, making the process of factoring and solving the polynomial more manageable.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a zero at x = c, then (x - c) is a factor of the polynomial. This concept is crucial in breaking down the polynomial into simpler components once a zero is identified, facilitating further factorization and solution finding.

Transcript

00:01 We want to find all of the real zeros of this polynomial 3x cubed plus 4x squared minus 13x plus 6.

00:11 And it asks to find all the real zeros using the rational zero theorem.

00:16 And what the rational zero theorem tells is that if we have these two values p, which is the factors of the constant term.

00:28 In this case the factors of positive 6 and q which is what we consider the factors of the leading term or the leading coefficient which is just 3 in this case we know that all possible rational zeros must be of the form p divided by q and so what we can do is find p and find q and give us what our possible zeros could be.

01:13 So we know p is the factors of the constant term, which we said is just positive six, and those are plus and minus one, plus it minus two, plus and minus three, and plus and minus six.

01:26 And q is just the factors of three, our leading term, which are plus and minus one, plus and minus three.

01:34 Okay, so now we can write, p is over q.

01:42 So p over q, notice that if we divide by plus minus one, we keep the same p's and q.

01:49 So we have plus minus one, plus minus two, plus minus three, plus minus six.

01:56 And then if we divide everything by three, we get plus minus one -third, plus minus two -thirds, plus minus three -thirds, which is plus minus one, and then plus minus six -thirds, which is just two.

02:08 So we don't have to include this again.

02:11 And so these are all of the possible p over q values.

02:15 And it is going to be a lot of work to check every single one.

02:19 But if we go ahead and check in order, once we find at least one, we know that we can just use synthetic division, factor it out, and then we should be left with a quadratic term that hopefully we'll be able to factor as well.

02:36 So let's start with x is equal to positive 1.

02:39 So x equals 1 and we can rewrite our polynomial 3x cubed plus 4 x squared minus 13x plus 6 and we want to solve this when this is equal to 0 and so if we go ahead and write our synthetic division we have the coefficients 3 4 negative 13 and 6 and we're going to use positive 1 so we bring down this 3 we end up with 3 times 1 which is 3 and then 4 plus 3 is 7, 7 times 1 is 7, and then negative 13 plus 7 is negative 6, and negative 6 times 1 is negative 6 again.

03:22 And we can see we are given a 0.

03:25 So we know that this is a 0.

03:27 So luckily on our first guess, x equals 1, we found that this is a 0...

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