5 55 6 Theorems about Roots of Polynomial

5. 5/5. 6 Theorems about Roots of Polynomial Equations Learning Goal • solve quadratic equations with real coefficients that have complex solutions

Factor Theorem If is a factor, then is a zero of the polynomial Remainder Theorem If you divide a polynomial then the remainder is ,

Rational Root Theorem is a factor of P(x) if and only if

Ex 1 Find all the zeros.

Rational Zero Theorem • If the polynomial P(x) has integer coefficients then every rational zero of P is of the form p/q where p is a factor of the constant and q is a factor of the leading coefficient. • 3 lists: Factors of p: Factors of q: Possible roots: p/q

Ex 2 List all possible rational roots. • P(x) = 9 x 6 – 5 x 3 + 27 Possible roots: p/q Factors of p: Factors of q: Final Possibilities Cancel out duplicates

Ex 3 List all possible rational roots and then find all the roots.

Ex 4 Use the rational zero theorem to factor: • P(x) = 3 x 3 – 4 x 2 – 13 x - 6 Final Possibilities Possible roots: p/q Use synthetic division to find one of the factors This means the factors are: Now factor the quadratic.

Ex 5 Use the rational zero theorem to factor: • P(x) = -x 3 – 3 x 2 + 13 x + 15 Use synthetic division to check This means the factors are: Now factor the quadratic. Final Possibilities

Ex 6 Find all the real zeros • P(x) = x 3 – 6 x + 4 Final Possibilities Use synthetic division to check Zeros: Factors: Use the quadratic formula.

Conjugate Root Theorem if then is a root, is also a root. AND if then is a root, is also a root. Recall, they are CONJUGATES!

Ex 7 A polynomial equation with rational coefficients has the following roots; find two additional roots. Ex 8 A polynomial equation with rational coefficients has the following roots; find two additional roots.

Ex 9 Find a 3 rd degree polynomial with rational coefficients that has the following roots:

Homework pg 322 #16 -18, 20, 24