Finding Zeros of a Polynomial Function Fundamental Theorem

Finding Zeros of a Polynomial Function

Fundamental Theorem of Algebra (FTA) every polynomial has at least one solution

Fundamental Theorem of Algebra (FTA) COROLLARY The degree (biggest exponent) = # of roots

These words mean the same thing • • Roots Zeros Solutions X intercepts • A factor is just x - #

Decarte’s Rule of Signs The number of times the sign changes in p(x)= possible number of positive roots Or 2 less, 4 less, 6 less, etc

Decarte’s Rule of Signs The number of times the sign changes in p(-x)= possible number of negative roots Or 2 less, 4 less, 6 less, etc

Decarte’s Rule of Signs Can sometimes narrow down which numbers to check Can also tell how many imaginary roots are possible. Degree – (# of + plus # -)

Ex with Decarte’s Rule • 4 x^3 – 7 x + 3 = p(x) • Signs in order [p(x)] + - + • There are 2 or 0 positive roots • • • P(-x)= 4(-x)^3 -7(-x)+3 =-4 x^3 + 7 x + 3 Signs in order for p(-x): - + + There is only 1 sign change We are guaranteed 1 negative root • CHECK NEGATIVE NUMBERS

Rational Zero Theorem P = all the numbers you can multiply to get the constant Q = all the numbers you can multiply to get the leading coefficient +- p/q = all POSSIBLE factors of your polynomial

Upper bound/lower bound • Will cover these 2 in 4 -5 • Tells us there will be no roots above # • Tells us there will be no roots below # • Uses synthetic division

Location Principal • Helps find fractional and irrational zeros • Uses synthetic division or graph

Factor Theorem When using synthetic division, if the remainder is 0, then the # you divided by is a root, zero, solution, x -intercept AND X – divisor Is a factor!

Put it together…

Finding zeros Location Principal Factor Thm Upper/Lower bounds thms Decarte’s Rule Rational Zero thm FTA and it’s corollary

Organizational Chart FTA Rational Zero Thm Decarte’s Rule of Signs Upper/Lower Bound Thm Factor Thm Solve Quadratics Location Principal

Let’s do an example • • 4 x^3 – 7 x +3 FTA– there is at least one root Corollary There are 3 roots Decarte’s rule 2 or 0 positive 1 negative 2 or 0 imaginary

4 x^3 – 7 x +3 continued • Rational Zero Thm • + or -, 1, 3, 1/2, 1/4, 3/2, 3/4 • We know that we are guaranteed 1 negative root • Start checking negative roots • While checking, notice the quotient– if all positive #’s, that’s the upper bound • While checking, notice the quotient– if signs alternate, that’s a lower bound

4 x^3 – 7 x +3 continued • While checking, notice the remainder • What does the factor thm say? • When we divide by – 3/2 the remainder is zero: -3/2 4 0 -7 3 -6 9 -3 4 -6 2 0

4 x^3 – 7 x +3 continued • • Quotient: 4 x^2 -6 x + 2 Factors as 2(x – 1)(2 x – 1) While checking consecutive integers, check for sign change in the remainder location principal • We have all 3 zeros • 1, ½, -3/2