Download Remainder Theorem and Factor Theorem

Unit 3 Day 9 - The Real Zeroes of
a Polynomial Function
We will use the Remainder Theorem
and Synthetic Division to determine
if a number is a root of a polynomial
And
I will determine if a number is a root
of a polynomial.
October 21, 2016
If r is a real zero of a polynomial function
f, then
• f(r) = 0
• r is an x-intercept of the graph of f
• r is a solution of the equation f(x) = 0
• r is also called a root of the equation
• (x – r) is a factor of the polynomial
• if you divide the polynomial by (x –r),
the remainder would be zero.
Remainder Theorem
Let f(x) be a polynomial function. If f(x) is
divided by x – c, then the remainder is f(c).
So one way to see if c is a factor of a
polynomial is to use long division and see if you
get a remainder.
Factor Theorem
• Let f(x) be a polynomial function. Then
x – c is a factor of f(x) if and only if
f(c) = 0
Example: Use the Factor Theorem to determine
whether (x – c) is a factor of f.
f(x) = x3 - 3x2 + x + 2 c = 2
f(2) = 23 - 3(2)2 + 2 + 2
f(2) = 8 - 12 + 2 + 2
f(2) = 0
2 is a factor
Example: Determine whether (x – c) is a factor of f.
If it is, write f in factored form, that is, write f in
the form f(x) = (x – c)(quotient).
f(x) = 4x3 – 3x2 – 8x + 4
c=3
There are two ways to determine if it is a factor:
1. Use the factor or remainder theorem
f(3)
f(3)
f(3)
f(3)
f(3)
f(3)
= 4(3)3 – 3(3)2 – 8(3) + 4
= 4(27) – 3(9) – 24 + 4
= 108 – 27 -20
= 108 – 47
= 61
≠ 0, so it is not a factor
Synthetic Division
• A shorthand method of dividing polynomials
• You divide the coefficients of the
polynomials, removing the variables and
exponents.
• It allows you to add throughout the process
instead of subtract, as you would do in
traditional long division.
Synthetic Division
Divide 4x3 – 3x2 – 8x + 4 by x – 3:
1. Make sure the powers are in decreasing
order.
2. List the coefficients in order (add a zero
for any missing powers of x)
3. Add a division symbol and dividing by ‘c’
3 4 -3 -8 4
Synthetic Division
Divide 4x3 – 3x2 – 8x + 4 by x – 3:
4. Add a horizontal line leaving room for work.
5. Drop the first coefficient below the
horizontal line.
6. Multiply that number by ‘c’. Whatever its
product, place it above the horizontal line
just below the second coefficient.
3 4 3 8 4
12
4
Synthetic Division
Divide 4x3 – 3x2 – 8x + 4 by x – 3:
7. Add the column of numbers, then put the
sum directly below the horizontal line.
8. Repeat the process until you run out of
columns to add.
3 4 3 8 4
12 27 57
4
9 19 61
The last number below the horizontal line is
always the remainder!
Example: Determine whether (x – c) is a factor of f.
If it is, write f in factored form, that is, write f in
the form f(x) = (x – c)(quotient).
f(x) = 4x4 – 15x2 – 4 c = 2
2 4 0  15 0  4
8
16 2
4
4 8
1 2

The remainder is 0, meaning that (x – 2) is a
factor.
The numbers below the horizontal line except the last
(remainder) are the coefficients of the Quotient.
The exponents of the variables of the quotient are all
reduced by 1.
f(x) = 4x4 – 15x2 – 4 c = 2
2 4 0  15 0  4
8
16 2
4
4 8
1 2

f(x) = (x – 2)(4x3 + 8x2 + x + 2)