4.4 Notes The Rational Root Theorem. 4.4 Notes To solve a polynomial equation, begin by getting the equation in standard form set equal to zero. Then.

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Presentation transcript:

4.4 Notes The Rational Root Theorem

4.4 Notes To solve a polynomial equation, begin by getting the equation in standard form set equal to zero. Then factor by one or more of the five methods. Set each factor equal to zero and solve. What if the polynomial won’t factor by one of the five methods?

4.4 Notes Determine if is a factor of.

4.4 Notes We know that is a factor of What does this mean? It means that will divide into without a remainder. Let’s divide it!

4.4 Notes

So becomes. And can be factored. So becomes. The roots are -4, -1, and 2.

4.4 Notes So if we can’t factor a polynomial, but we know a factor of the polynomial, we can divide the factor into the polynomial. This is called “depressing the polynomial.” The quotient may be factorable by one of the five methods of factoring. If we had a way to find a factor of the polynomial then we could use this process whenever the polynomial isn’t factorable.

4.4 Notes We do have a way! It’s called the Rational Root Theorem. This theorem predicts all possible rational roots of a polynomial.

4.4 Notes The Rational Root Theorem – Let represent a polynomial equation of degree n with integral coefficients. If a rational number, where p and q have no common factors, is a root of the equation, then p is a factor of a n and q is a factor of a o.

4.4 Notes “ is a possible root of P(x) if p is a factor of a n and q is a factor of a o.”

4.4 Notes Making a list of possible rational roots: 1.Get equation in standard form. 2.Identify a n (the constant) and make a list of integral factors of a n. 3.Identify a o (the coefficient of the term with the greatest exponent) and make a list of integral factors of a o. 4.Make a list of all possible combinations of fractions of factors of a n over factors of a o.

4.4 Notes 1. a n = 3: a o = 1: possible roots:

4.4 Notes 2. a n = 1: a o = 2: possible roots:

4.4 Notes 3. a n = -6: a o = 3: possible roots:

4.4 Notes Turn to p. 233 in your book. Answer 5 and 6. Just list all possible rational roots.

4.4 Notes 5. 6.