5.4 – Apply the Remainder and Factor Theorems

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) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5

5.4 – Apply the Remainder and Factor Theorems

Example 1b:

Divide f(x) = x3 +3x2 – 7 by x2 – x – 2

5.4 – Apply the Remainder and Factor Theorems

Example 2:

Divide f(x) = x3 +5x2 – 7x + 2 by x – 2

5.4 – Apply the Remainder and Factor Theorems

Example 2b:

Divide f(x) = 3x3 + 17x2 + 21x – 11 by x + 3

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.

5.4 – Apply the Remainder and Factor Theorems

Example 3:

Divide f(x) = 2x3 + x2 – 8x + 5 by x + 3 using synthetic division

5.4 – Apply the Remainder and Factor Theorems

Example 3b:

Divide f(x) = 2x3 + 9x2 + 14x + 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) = 3x3 – 4x2 – 28x – 16 completely given that x + 2 is a factor.

5.4 – Apply the Remainder and Factor Theorems

Example 4b:

Factor f(x) = 2x3 – 11x2 + 3x + 36 completely given that x – 3 is a factor.

5.4 – Apply the Remainder and Factor Theorems

Example 5:

One zero of f(x) = x3 – 2x2 – 23x + 60 is x = 3. What is another zero of f?