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5.4 – Apply the Remainder and Factor Theorems Divide 247 / 3 8829 / 8

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5.4 – Apply the Remainder and Factor Theorems When you divide a polynomial f(x) by a divsor d(x), you get a quotient polynomial q(x) and a remainder polynomial r(x). f(x) / d(x) = q(x) + r(x)/d(x) The degree of the remainder must be less than the degree of the divisor. One way to divide polynomials is called polynomial long division.

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5.4 – Apply the Remainder and Factor Theorems Example 1: Divide f(x) = 3x 4 – 5x 3 + 4x – 6 by x 2 – 3x + 5

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5.4 – Apply the Remainder and Factor Theorems Example 1b: Divide f(x) = x 3 +3x 2 – 7 by x 2 – x – 2

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5.4 – Apply the Remainder and Factor Theorems Example 2: Divide f(x) = x 3 +5x 2 – 7x + 2 by x – 2

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5.4 – Apply the Remainder and Factor Theorems Example 2b: Divide f(x) = 3x 3 + 17x 2 + 21x – 11 by x + 3

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5.4 – Apply the Remainder and Factor Theorems Synthetic division can be used to divide any polynomial by a divisor of the form x – k.

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5.4 – Apply the Remainder and Factor Theorems Example 3: Divide f(x) = 2x 3 + x 2 – 8x + 5 by x + 3 using synthetic division

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5.4 – Apply the Remainder and Factor Theorems Example 3b: Divide f(x) = 2x 3 + 9x 2 + 14x + 5 by x – 3 using synthetic division

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5.4 – Apply the Remainder and Factor Theorems

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Example 4: Factor f(x) = 3x 3 – 4x 2 – 28x – 16 completely given that x + 2 is a factor.

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5.4 – Apply the Remainder and Factor Theorems Example 4b: Factor f(x) = 2x 3 – 11x 2 + 3x + 36 completely given that x – 3 is a factor.

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5.4 – Apply the Remainder and Factor Theorems Example 5: One zero of f(x) = x 3 – 2x 2 – 23x + 60 is x = 3. What is another zero of f?

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