# 7.4 Remainder and Factor Theorems 7.5 Roots and Zeros

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7.4 Remainder and Factor Theorems 7.5 Roots and Zeros
Algebra II w/ trig

2 Methods for Polynomial Division can be used to find a quotient and remainder:
Long division: will work for divisors of any degree Synthetic Division: is quicker, but only will work for divisors of the form x+k Long Division: synthetic division:

Or you can use Synthetic Substitution:
If f(x) = -16t2 + 74t + 5 Find f(3):

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

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.

B. Given a polynomial and one of its factors, find the remaining factors of the polynomials. 1.

2.

3.

4.

7.5 Roots and Zeros FUNDAMENTAL 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.

I. Given a function and one of its zeros, find the remaining zeros of the functions.

B.

C.

D.

E.

II. Write a polynomial equation with the given roots.
A. 6, 2i B. 1, 1+i

C. -2, 2+3i D.