December 09, 2016
2.3 Polynomial Long Division
2
1.) Divide f(x) = 2x -7x+9 by (x-2) and show a check.
Check:
2.)
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3.)
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x-3 divided evenly into 2x3 - 9x2 + 11x - 6
so x-3 is a factor of this polynomial.
Divide 2x2 - 7x + 9 by x - 2
Long Division
Synthetic Division
December 09, 2016
Synthetic Division - can only be used when dividing
by a linear binomial with leading coefficient of 1 (ie. x+4,
x-5, etc.).
Divide 2x2 - 7x + 9 by x - 2
Synthetic Division
Long Division
Use synthetic division.
December 09, 2016
2.)
Synthetic Division Group Activity
1.) Each member of the group writes a polynomial
2.) Choose a factor x - a to divide into that polynomial.
3.) Divide f(x) by x - a using synthetic division.
4.) Evaluate f(a)
5.) Compare all examples in the group.
6.) Do you notice anything?
7.) Why does this happen? Explain.
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Factor Theorem
If (x - c) divides into a polynomial evenly
with no remainder,
then (x - c) is a factor of the polynomial
C would be a zero (x-intercept) of the
polynomial function.
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Remainder Theorem
For f(x) divided by x - a,
the remainder is the value of f(a).
Ex. If f(x) = 2x2 - 7x + 9 and
(2x2 - 7x + 9) ÷ (x - 2) = 2x - 3 with remainder 3
then f(2) = 3
because f(x) = (2x - 3) (x - 2) + 3
and f(2) = (2 2 - 3) (2 - 2) + 3
so
f(2) = 3
1a.) Use polynomial long division to divide:
(9x4 + 6x3 + 4x + 4) ÷ (3x2 +2x + 2)
b.) Show a check.
2.) f(x) = 4x3 - 3x - 2
a.) Use synthetic division to divide:
(4x3 - 3x - 2) ÷ (x + 1)
b.) Show a check.
c.) Is (x + 1) a factor of (4x3 - 3x - 2)? How do you know?
d.) What is f(-1)? (Do not substitute -1 into f(x))
December 09, 2016
1a.) Use polynomial long division to divide:
(9x4 + 6x3 + 4x + 4) ÷ (3x2 +2x + 2)
b.) Show a check.
Check:
2.) f(x) = 4x3 - 3x - 2
a.) Use synthetic division to divide:
(4x3 - 3x - 2) ÷ (x + 1)
b.) Show a check.
c.) Is (x + 1) a factor of (4x3 - 3x - 2)? How do you know?
d.) What is f(-1)? (Do not substitute -1 into f(x))
Check:
c) x+l is not a factor because there
is a remainder.
d) f(-1) = -3 which is the remainder
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4
2
2
1.) Divide: (2x - 3x + 7x - 8) ÷ (x + x - 3)
3
2
2.) Given f(x) = -x + 2x + 4,
a.) use synthetic division to find f(-2).
b.) Is x = -2 an x-intercept of f(x)? How do you know?
a.) f(-2) = 20
b.) No. The y-coordinate is not 0.
(There is a remainder when you divide by x+2.)
Solve:
x3 - 3x + 2 = 0
given that (x - 1) is a factor.
Use synthetic division to divide by x-1
Suppose I tell you that (x+2) is a factor or -2 is one
solution.
Can you find the others? How?
then factor the resulting trinomial
How can we find a factor to start with?
We can use the rational zero test.
Knowing one factor helps you to find others
December 09, 2016
Factor completely.
The Rational Zero Test (Rational Root Theorem)
Rational Zero Test
If a polynomial function has integer
coefficients, every root is of the form p/q
where
p is a factor of the constant term and
q is a factor of the leading coefficient
p/q is in lowest terms
Possible rational zeros = factors of the constant term
factors of leading coefficient
You can find the actual rational zeros using synthetic
division to test the possible zeros.
This helps you to find one zero to start with.
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List the possible rational zeros.
Do not find the actual zeros.
f(x) = 2x2 - 3x + 6
Example:
Find the rational zeros of f(x) = x3 - 2x2 - x + 2
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Find all rational zeros and factor the polynomial completely.
f(x) = x3 - 5x2 + 2x + 8
2, 4, -1
f(x) = 2x3 + 3x2 - 8x + 3
Find all real zeros of the function and factor
the polynomial completely.
f(x) = 5x3 - 9x2 - 12x - 2
Does not factor. Use
quad. formula to find zeros.
December 09, 2016
Find all real zeros of the function and factor
the polynomial completely.
f(x) = x3 - 4x2 - 3x + 14
Find all real zeros of the function and factor
the polynomial completely.
f(x) = x3 - 4x2 - 3x + 14
December 09, 2016
Answers to Wksht "Rational Zero Test"
List all combinations
omit duplicates