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**7.4 Remainder and Factor Theorems 7.5 Roots and Zeros**

Algebra II w/ trig

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**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:

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**Or you can use Synthetic Substitution:**

If f(x) = -16t2 + 74t + 5 Find f(3):

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**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

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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.

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

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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.

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

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B.

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C.

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D.

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E.

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**II. Write a polynomial equation with the given roots.**

A. 6, 2i B. 1, 1+i

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C. -2, 2+3i D.

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