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 CHECK NEGATIVE NUMBERS 4x^3 – 7x + 3 = p(x) Signs in order [p(x)] + - + There are 2 or 0 positive roots P(-x)= 4(-x)^3-7(-x)+3 =-4x^3 + 7x + 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

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

Let’s do an example 4x^3 – 7x +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

4x^3 – 7x +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

4x^3 – 7x +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

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