Presentation on theme: "7.4 Remainder and Factor Theorems 7.5 Roots and Zeros"— Presentation transcript:
1 7.4 Remainder and Factor Theorems 7.5 Roots and Zeros Algebra II w/ trig
2 2 Methods for Polynomial Division can be used to find a quotient and remainder: Long division: will work for divisors of any degreeSynthetic Division: is quicker, but only will work for divisors of the form x+kLong Division: synthetic division:
3 Or you can use Synthetic Substitution: If f(x) = -16t2 + 74t + 5Find f(3):
4 I. REMAINDER THEOREM: If a polynomial f(x) is divided by (x-c), the remainder is f(c). A. Using synthetic substitution(use when degree is greater than 2) to find f(-3) :if
5 II. FACTOR THEOREM: A polynomial f(x) has a factor (x-k) if and only if f(k)=0, so if the remainder is zero.A. Show that (x+5) is a factor of Then find the remaining factor(s) of the polynomial.
6 B. Given a polynomial and one of its factors, find the remaining factors of the polynomials. 1.
10 7.5 Roots and ZerosFUNDAMENTAL THEOREM OF ALGEBRA: If f(x) is a polynomial with positive degree, then f(x) has at least one root.In general:Degree = # of solutions, roots, zeros (but sometimes the same solution can happen more than one (double root - (x+2)2 ; x = -2)Imaginary solutions always occur in pairs: If (a+bi) is a solution, then automatically we have (a – bi) is a solution as well.
11 I. Given a function and one of its zeros, find the remaining zeros of the functions.