2.3: Polynomial Division

1
2.3 Polynomial Division

• Objectives
• To divide polynomials using long and synthetic
  division
• To apply the Factor and Remainder Theorems to
  find real zeros of polynomial functions

2
Vocabulary

• As a class, use your vast mathematical knowledge
  to define each of these words without the aid of
  your textbook.

Quotient Remainder
Dividend Divisor
Divides Evenly Factor
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Exercise 1

• Use long division to divide 5 into 3462.

-
-
-
4
Exercise 1

• Use long division to divide 5 into 3462.

Quotient
Divisor
Dividend
-
-
-
Remainder
5
Exercise 1

• Use long division to divide 5 into 3462.

Dividend
Remainder
Divisor
Divisor
Quotient
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Remainders

• If you are lucky enough to get a remainder of
  zero when dividing, then the divisor divides
  evenly into the dividend.
• This means that the divisor is a factor of the
  dividend.
• For example, when dividing 3 into 192, the
  remainder is 0. Therefore, 3 is a factor of 192.

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Dividing Polynomials

• Dividing polynomials works just like long
  division. In fact, it is called long division!
• Before you start dividing
• Make sure the divisor and dividend are in
  standard form (highest to lowest powers).
• If your polynomial is missing a term, add it in
  with a coefficient of 0 as a place holder.

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Exercise 2

• Divide x 1 into x2 3x 5
• Line up the first term of the quotient with the
  term of the dividend with the same degree.

How many times does x go into x2?
Multiply x by x 1
-
-
Multiply 2 by x 1
-
-
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Exercise 2

• Divide x 1 into x2 3x 5

Quotient
Dividend
-
-
-
-
Divisor
Remainder
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Exercise 2

• Divide x 1 into x2 3x 5

Dividend
Remainder
Divisor
Quotient
Divisor
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Exercise 3

• Divide 6x3 16x2 17x 6 by 3x 2

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Exercise 4

• Use long division to divide x4 10x2 2x 3 by
  x 3

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Synthetic Division

• When your divisor is of the form x - k, where k
  is a constant, then you can perform the division
  quicker and easier using just the coefficients of
  the dividend.
• This is called fake division. I mean, synthetic
  division.

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Synthetic Division

• Synthetic Division (of a Cubic Polynomial)
• To divide ax3 bx2 cx d by x k, use the
  following pattern.

Add terms
k
a
b
c
d
ka
Multiply by k
a
Remainder
Coefficients of Quotient (in decreasing order)
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Synthetic Division

• Synthetic Division (of a Cubic Polynomial)
• To divide ax3 bx2 cx d by x k, use the
  following pattern.
• Important Note You are always adding columns
  using synthetic division, whereas you subtracted
  columns in long division.

Add terms
k
a
b
c
d
ka
Multiply by k
a
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Synthetic Division

• Synthetic Division (of a Cubic Polynomial)
• To divide ax3 bx2 cx d by x k, use the
  following pattern.
• Important Note k can be positive or negative.
  If you divide by x 2, then k -2 because x 2
  x (-2).

Add terms
k
a
b
c
d
ka
Multiply by k
a
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Synthetic Division

• Synthetic Division (of a Cubic Polynomial)
• To divide ax3 bx2 cx d by x k, use the
  following pattern.
• Important Note Add a coefficient of zero for
  any missing terms!

Add terms
k
a
b
c
d
ka
Multiply by k
a
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Exercise 5

• Use synthetic division to divide x4 10x2 2x
  3 by x 3

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Exercise 6

• Evaluate f (3) for f (x) x4 10x2 2x 3.

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Remainder Theorem

• If a polynomial f (x) is divided by x k, the
  remainder is r f (k).
• This means that you could use synthetic division
  to evaluate f (5) or f (-2). Your answer will be
  the remainder.

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Exercise 7

• Divide 2x3 9x2 4x 5 by x 3 using
  synthetic division.

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Exercise 8

• Use synthetic division to divide
  f(x) 2x3 11x2 3x 36 by x 3.
• Since the remainder is zero when dividing f(x) by
  x 3, we can write
• This means that x 3 is a factor of f(x).

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Factor Theorem

• A polynomial f(x) has a factor x k if and only
  if f(k) 0.
• This theorem can be used to help factor/solve a
  polynomial function if you already know one of
  the factors.

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Exercise 9

• Factor f(x) 2x3 11x2 3x 36 given that x
  3 is one factor of f(x). Then find the zeros of
  f(x).

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Exercise 10

• Given that x 4 is a factor of x3 6x2 5x
  12, rewrite x3 6x2 5x 12 as a product of
  two polynomials.

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Exercise 11

• Find the other zeros of f(x) 10x3 81x2 71x
  42 given that f(7) 0.

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Rational Zero Test we use this to find the
rational zeros for a polynomial f(x). It says
that if f(x) is a polynomial of the form
Then the rational zeros of f(x) will be of the
form
Where p factor of the constant q
factor of leading coefficient
Rational zero
Possible rational zeros factors of the
constant term___ factors of the leading
coefficient

• Keep in mind that a polynomial can have rational
  zeros, irrational zeros and complex zeros.

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Ex 1 Find all of the possible rational zeros of
f(x)
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Ex 2 Find the rational zeros of
Lets start by listing all of the possible
rational zeros, then we will use synthetic
division to test out the zeros 1. Start with a
list of factors of -6 (the constant term) p
2. Next create a list of factors of 1
(leading coefficient) q 3. Now list your
possible rational zeros p/q Testing all of
those possibilities could take a while so lets
use the graph of f(x) to locate good
possibilities for zeros.
Use your trace button!
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Ex 2 continued Find all of the rational zeros
of the function
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Ex 3 Find all the real zeros of
p Factors of 3 q Factors of 2 Candidates
for rational zeros p/q Lets look at the
graph Which looks worth trying? Now use
synthetic division to test them out.
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Homework

• Dividing Polynomials Worksheet
• Page 127-128
• 36,38, 49-59 odd