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

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

Key Concepts

Real Zeros

Real zeros of a polynomial are the x-values where the function crosses or touches the x-axis. These zeros represent the solutions to the polynomial equation. In the context of polynomial factorization and the application of the Rational Zero Theorem, identifying the real zeros is essential for completely solving the equation and understanding the behavior of the polynomial function.

Synthetic Division

Synthetic Division is a simplified method of dividing a polynomial by a linear divisor of the form (x - c). It is particularly useful for testing potential zeros determined by the Rational Zero Theorem. By using synthetic division, one can efficiently determine whether a candidate zero actually makes the polynomial zero and obtain the coefficients of the reduced polynomial for further analysis.

Rational Zero Theorem

The Rational Zero Theorem is a tool used in algebra to identify all possible rational roots of a polynomial with integer coefficients. It states that any potential rational zero of the polynomial is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem helps narrow down the list of candidate roots that need to be tested when solving polynomial equations.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a value of zero at x = c (i.e., f(c) = 0), then (x - c) is a factor of the polynomial. This theorem is closely related to the Rational Zero Theorem because once a possible rational zero is found, it can be tested; if it yields zero, it confirms that the corresponding linear factor divides the polynomial.

Transcript

00:01 So we want to use the rational zero theorem to find all of the real zeros of the polynomial x cubed minus 3x squared minus 3x minus 4.

00:14 And so what the rational zero theorem tells us is that if we consider p as the factors of the constant term, and then q as the factors of the leading coefficient, then all possible rational zeros are of the form p divided by q.

00:45 So that means that we just have to find all the factors of negative 4 and all the factors of 1 for this problem, and then divide the 2, and that'll give us the list of all possible zeros for this polynomial.

00:58 And so we know that our constant is negative 4, and our leading term is this 1, as i said.

01:05 So we know p is just the factors of negative 4, which is just plus minus 1, plus minus 2, and plus minus 4.

01:12 And so q is just also the factors of the coefficient 1 is just plus minus 1.

01:21 And so since we're using the form p over q, we know that since q is just 1, a positive and negative 1, that all the p values are just the possible values of our 0s.

01:35 So we can do a quick check through all of these values.

01:41 And it's pretty easy to see that x equals 1 won't work because we have one positive coefficient of positive one and then the rest are negative so we would not get a zero from that and let's go ahead and start writing so we can do this more rigorously so just by looking at this i've already ruled out x equals 1 is not so 0 so let's try negative 1 maybe that'll work so if you plug in our coefficients let's go ahead and actually write x equals negative 1 so if we go ahead and use our guess and do synthetic division we can go ahead and check and see if it's a zero and if it is we'll automatically get its factorization as well so if you go ahead and try this we can see we have doing the synthetic division, i'm sorry, oops, so one, and we'll see that this is not a zero either.

02:53 So x is equal to negative one is not a zero.

02:58 Let's try x is equal to two.

03:01 So x is equal to two.

03:06 Let's see, so we have two, bringing down this one, we have two, negative one, negative two, negative five, negative ten, and negative fourteen, which is also not a zero.

03:21 Okay, so it's going to take a little while, but we can go ahead and keep trying.

03:25 Let's try x is equal to negative 2, 1, negative 3, negative 4, and we'll see we get negative, sorry, we have 1, negative 2, negative 5, we get positive 10, 7, negative 14, which gives us negative 18...

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