5.5 – Dividing Polynomials

Divide

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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.

5.4 – Apply the Remainder and Factor Theorems

Example 1:Divide f(x) = x3 +5x2 – 7x + 2 by x – 2

What is the quotient and remainder??

5.4 – Apply the Remainder and Factor Theorems

Example 1b:Divide f(x) = x3 +3x2 – 7 by x2 – x – 2

What is the quotient and remainder????

5.4 – Apply the Remainder and Factor Theorems

Example 1c:Divide f(x) = 3x4 – 5x3 + 4x – 6 by x2 – 3x + 5

What is the quotient and remainder???

5.4 – Apply the Remainder and Factor Theorems

Example 1d:Divide f(x) = 3x3 + 17x2 + 21x – 11 by x + 3

What is the quotient and remainder???

5.4 – Apply the Remainder and Factor Theorems

Example 2:Is x2 + 1 a factor of 3x4 – 4x3 +12x2 + 5

How do you know??????????

5.4 – Apply the Remainder and Factor Theorems

Example 2b:Is x – 2 a factor of P(x) = x5 – 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) = 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?

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) = x5 – 2x3 – x2 + 2, what is the

remainder when P(x) is divided by x – 3?

5.4 – Apply the Remainder and Factor Theorems

Example 6b:Given that P(x) = x5 – 3x4 - 28x3+ 5x + 20, what is the remainder when P(x) is divided by x + 4 ?