5 5 Dividing Polynomials Divide 247 3 8829

5. 5 – Dividing Polynomials Divide 247 / 3 8829 / 8

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.

5. 4 – Apply the Remainder and Factor Theorems Example 1: Divide f(x) = x 3 +5 x 2 – 7 x + 2 by x – 2 What is the quotient and remainder? ?

5. 4 – Apply the Remainder and Factor Theorems Example 1 b: Divide f(x) = x 3 +3 x 2 – 7 by x 2 – x – 2 What is the quotient and remainder? ?

5. 4 – Apply the Remainder and Factor Theorems Example 1 c: Divide f(x) = 3 x 4 – 5 x 3 + 4 x – 6 by x 2 – 3 x + 5 What is the quotient and remainder? ? ?

5. 4 – Apply the Remainder and Factor Theorems Example 1 d: Divide f(x) = 3 x 3 + 17 x 2 + 21 x – 11 by x + 3 What is the quotient and remainder? ? ?

5. 4 – Apply the Remainder and Factor Theorems Example 2: Is x 2 + 1 a factor of 3 x 4 – 4 x 3 +12 x 2 + 5 How do you know? ? ? ? ?

5. 4 – Apply the Remainder and Factor Theorems Example 2 b: Is x – 2 a factor of P(x) = x 5 – 32? ? If it is, write P(x) as a product of two factors. How do you know? ? ? ? ?

5. 4 – Apply the Remainder and Factor Theorems Synthetic division simplifies the long-division process for dividing by a linear expression x – a. To use synthetic division, write the coefficients (including zeros) of the polynomial in standard form. Omit all variables and exponents. For the divisor, reverse the sign (use a). This allows you to add instead of subtract throughout the process.

5. 4 – Apply the Remainder and Factor Theorems Example 3: Divide f(x) = 2 x 3 + x 2 – 8 x + 5 by x + 3 using synthetic division

5. 4 – Apply the Remainder and Factor Theorems Example 3 b: Divide f(x) = 2 x 3 + 9 x 2 + 14 x + 5 by x – 3 using synthetic division

5. 4 – Apply the Remainder and Factor Theorems

5. 4 – Apply the Remainder and Factor Theorems Example 4: Factor f(x) = 3 x 3 – 4 x 2 – 28 x – 16 completely given that x + 2 is a factor.

5. 4 – Apply the Remainder and Factor Theorems Example 4 b: Factor f(x) = 2 x 3 – 11 x 2 + 3 x + 36 completely given that x – 3 is a factor.

5. 4 – Apply the Remainder and Factor Theorems Example 5: One zero of f(x) = x 3 – 2 x 2 – 23 x + 60 is x = 3. What is another zero of f?

5. 4 – Apply the Remainder and Factor Theorems The Remainder Theorem provides a quick way to find the remainder of a polynomial long division problem.

5. 4 – Apply the Remainder and Factor Theorems Example 6: Given that P(x) = x 5 – 2 x 3 – x 2 + 2, what is the remainder when P(x) is divided by x – 3?

5. 4 – Apply the Remainder and Factor Theorems Example 6 b: Given that P(x) = x 5 – 3 x 4 - 28 x 3+ 5 x + 20, what is the remainder when P(x) is divided by x + 4 ?