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5.5 Apply the Remainder and Factor Theorems What you should learn: Goal1 Divide polynomials and relate the result to the remainder theorem and the factor theorem. 5.5 The Remainder and Factor Theorem a) using Long Division b) Synthetic Division Goal2 Factoring using the Synthetic Method Goal3 Finding the other ZEROs when given one of them. A1.1.5

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Divide using the long division ex) x ( ) 6.5 The Remainder and Factor Theorem

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Divide using the long division with Missing Terms ex) - ( )

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Synthetic Division To divide a polynomial by x - c 1. Arrange polynomials in descending powers, with a 0 coefficient for any missing term. 2. Write c for the divisor, x – c. To the right, write the coefficients of the dividend

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3. Write the leading coefficient of the dividend on the bottom row. 4. Multiply c (in this case, 3) times the value just written on the bottom row. Write the product in the next column in the 2 nd row

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5. Add the values in the new column, writing the sum in the bottom row. 6. Repeat this series of multiplications and additions until all columns are filled in add 7 21 add 16

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7. Use the numbers in the last row to write the quotient and remainder in fractional form. The degree of the first term of the quotient is one less than the degree of the first term of the dividend. The final value in this row is the remainder add

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Synthetic Division To divide a polynomial by x - c Example 1)

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Synthetic Division To divide a polynomial by x - c Example 2)

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Factoring a Polynomial Example 1) given that f(-3) = multiply Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x). (x + 3)

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Factoring a Polynomial Example 2) given that f(2) = multiply Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x). (x - 2)

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Reflection on the Section If f(x) is a polynomial that has x – a as a factor, what do you know about the value of f(a)? assignment

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5.6 Finding Rational Zeros What you should learn: Goal1 Find the rational zeros of a polynomial. 5.6 Finding Rational Zeros L1.2.1

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Find the rational zeros of 5.6 Finding Rational Zeros The Rational Zero Theorem solution List the possible rational zeros. The leading coefficient is 1 and the constant term is -12. So, the possible rational zeros are:

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Find the Rational Zeros of 5.6 Finding Rational Zeros solution List the possible rational zeros. The leading coefficient is 2 and the constant term is 30. So, the possible rational zeros are: Example 1) Notice that we dont write the same numbers twice

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Example 2) Use Synthetic Division to decide which of the following are zeros of the function 1, -1, 2, -2 x = -2, Finding Rational Zeros

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Example 3) Find all the REAL Zeros of the function. x = -2, -3, Finding Rational Zeros

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Example 4) Find all the Real Zeros of the function Finding Rational Zeros

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x = 2, Finding Rational Zeros

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Reflection on the Section How can you use the graph of a polynomial function to help determine its real roots? assignment 5.6 Finding Rational Zeros

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5.7 Apply the Fundamental Theorem of Algebra What you should learn: Goal1 Use the fundamental theorem of algebra to determine the number of zeros of a polynomial function. 5.7 Using the Fundamental Theorem of Algebra THE FUNDEMENTAL THEOREM OF ALGEBRA If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers. L2.1.6

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Example 1) Find all the ZEROs of the polynomial function. x = -5, -3, Using the Fundamental Theorem of Algebra

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Example 1) Decide whether the given x-value is a zero of the function., x = -5 So, Yes the given x-value is a zero of the function Using the Fundamental Theorem of Algebra

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Example 1) Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1. -4, 1, Using the Fundamental Theorem of Algebra

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QUADRATIC FORMULA

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Example ) Find ALL the ZEROs of the polynomial function. x = x = -.732

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Example #24) Find ALL the ZEROs of the polynomial function. Doesnt FCTPOLY…Now what?

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Example ) Find ALL the ZEROs of the polynomial function.

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Example ) Find ALL the ZEROs of the polynomial function Graph this one….find one of the zeros..

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Reflection on the Section How can you tell from the factored form of a polynomial function whether the function has a repeated zero? assignment At least one of the factors will occur more than once.

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