Download Remainder and Factor Theorems

Remainder and Factor
Theorems
Divide f(x) = 4x2 – 3x + 6 by x – 2.
Long Division
What is f(2)?
Synthetic Division
In the previous problem, the value of f(2)
was the same as the remainder when
dividing by x – 2.
This illustrates the Remainder Theorem.
• When synthetic division is used to
evaluate a function, it is called synthetic
substitution.
• It is convenient when the degree of the
polynomial is greater than 2.
• When you divide a polynomial by one of its
binomial factors, the result is called a
depressed polynomial.
• If the remainder is 0 after completing
division, then x – a is a binomial factor of
the polynomial, and a is a zero. This is a
special case of the Remainder Theorem,
called the Factor Theorem.
• To show that a binomial is a factor of a
polynomial, we want to show that the
remainder is 0 after division.
• We can then find the remaining zeros or
roots of the polynomial using quadratic
methods.
Example: Show that x – 3 is a factor of x3 + 4x2 – 15x – 18.
Then find the remaining factors.
Try: Show that x + 3 is a factor of x3 + 6x2 – x – 30. Then
find the remaining factors.
Factoring
Grouping, Special Cases
• If you have 4 or more terms, you group
terms together so that they will have a
common factor.
• Factor each group separately.
• You will then be able to factor over the
entire expression.
• Rewrite.
Example: Factor x3 + 5x2 – 2x – 10
Example: Factor x3 – 4x2 + 3x – 12
Example: Factor 3x3 + 2xy – 15x2 – 10y
Difference of Two Squares
You may remember seeing expressions like this
one when you worked with multiplying algebraic
expressions. Do you remember ...
If you remember this fact, then you already know that
the factors of
are
and
Remember:
An algebraic term is a perfect square when the numerical coefficient
(the number in front of the variables) is a perfect square and the
exponents of each of the variables are even numbers.
Factor x2 - 9
Factor 9y2 - 16y6
Sum of Two Cubes
Example: x3 + 27
Difference of Two Cubes
Example: x3 – 64
What about x6 – y6 ?