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Long and Synthetic Division of Polynomials: Quotient and Remainder, Lecture notes of Algebra

Examples and explanations of long and synthetic division of polynomials, including the quotient and remainder. It covers dividing polynomials by binomials and using synthetic division as a shortcut.

What you will learn

  • What is the quotient and remainder when a polynomial is divided by a binomial using long division?
  • What is the difference between long division and synthetic division of polynomials?
  • How does synthetic division save time when dividing polynomials?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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polynomial equations & inequalities
MHF4U: Advanced Functions
Long and Synthetic Division of Polynomials
J. Garvin
Slide 1/12
polynomial equations & inequalities
Long Division
Recall...
What is the quotient and remainder when 357 is divided by
8?
44
8357
320
37
32
5
The quotient is 44 and the remainder is 5.
J. Garvin Long and Synthetic Division of Polynomials
Slide 2/12
polynomial equations & inequalities
Polynomial Division
Polynomials can be divided by other binomials in the same
way that rational numbers are divided by other rationals.
When doing long division with polynomials, each term in the
quotient should have a product with the divisor that is equal
in value to the current remainder.
It is important to insert terms with zero-coefficients as
required.
J. Garvin Long and Synthetic Division of Polynomials
Slide 3/12
polynomial equations & inequalities
Polynomial Division
Example
What are the quotient and remainder when
f(x) = x32x2+ 4x1 is divided by x2.
x2+ 0x+ 4
x2x32x2+ 4x1
x3+ 2x2
0x2+ 4x
0x2+ 0x
4x1
4x+ 8
7
The quotient is x2+ 4 and the remainder is 7.
J. Garvin Long and Synthetic Division of Polynomials
Slide 4/12
polynomial equations & inequalities
Polynomial Division
Example
What are the quotient and remainder when
f(x) = 3x35x+ 4 is divided by x+ 3.
3x29x+ 22
x+ 33x3+ 0x25x+ 4
3x39x2
9x25x
9x2+ 27x
22x+ 4
22x66
62
The quotient is 3x29x+ 22 and the remainder is 62.
J. Garvin Long and Synthetic Division of Polynomials
Slide 5/12
polynomial equations & inequalities
Polynomial Division
An alternate way of expressing the quotient and remainder
when a polynomial, P(x), is divided by a binomial, ax b, is
in the form P(x) = (ax b)Q(x) + R, where Q(x) is the
quotient and Ris the remainder.
Dividing this through by ax bresults in another form,
P(x)
ax b=Q(x) + R
ax b.
Thus, the last example could be rewritten as either
3x35x+ 4 = (x+ 3)(3x29x+ 22) 62, or as
3x35x+ 4
x+ 3 = 3x29x+ 22 62
x+ 3.
You should be familiar with these alternate forms, although
you may choose to use one of the three to express your
answers.
J. Garvin Long and Synthetic Division of Polynomials
Slide 6/12
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p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

MHF4U: Advanced Functions

Long and Synthetic Division of Polynomials

J. Garvin

Slide 1/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Long Division

Recall... What is the quotient and remainder when 357 is divided by 8? 44 8

The quotient is 44 and the remainder is 5.

J. Garvin — Long and Synthetic Division of PolynomialsSlide 2/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Polynomial Division

Polynomials can be divided by other binomials in the same way that rational numbers are divided by other rationals.

When doing long division with polynomials, each term in the quotient should have a product with the divisor that is equal in value to the current remainder.

It is important to insert terms with zero-coefficients as required.

J. Garvin — Long and Synthetic Division of Polynomials Slide 3/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Polynomial Division

Example What are the quotient and remainder when f (x) = x^3 − 2 x^2 + 4x − 1 is divided by x − 2.

x^2 + 0x + 4 x − 2

x^3 − 2 x^2 + 4x − 1 − x^3 + 2x^2 0 x^2 + 4x 0 x^2 + 0x 4 x − 1 − 4 x + 8 7

The quotient is x^2 + 4 and the remainder is 7. J. Garvin — Long and Synthetic Division of Polynomials Slide 4/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Polynomial Division

Example What are the quotient and remainder when f (x) = 3x^3 − 5 x + 4 is divided by x + 3.

3 x^2 − 9 x + 22 x + 3

3 x^3 + 0x^2 − 5 x + 4 − 3 x^3 − 9 x^2 − 9 x^2 − 5 x 9 x^2 + 27x 22 x + 4 − 22 x − 66 − 62

The quotient is 3x^2 − 9 x + 22 and the remainder is −62. J. Garvin — Long and Synthetic Division of PolynomialsSlide 5/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Polynomial Division

An alternate way of expressing the quotient and remainder when a polynomial, P(x), is divided by a binomial, ax − b, is in the form P(x) = (ax − b)Q(x) + R, where Q(x) is the quotient and R is the remainder. Dividing this through by ax − b results in another form, P(x) ax − b = Q(x) +

R

ax − b

Thus, the last example could be rewritten as either 3 x^3 − 5 x + 4 = (x + 3)(3x^2 − 9 x + 22) − 62, or as 3 x^3 − 5 x + 4 x + 3

= 3x^2 − 9 x + 22 −

x + 3

You should be familiar with these alternate forms, although you may choose to use one of the three to express your answers. J. Garvin — Long and Synthetic Division of PolynomialsSlide 6/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Synthetic Division

Long division can be tedious to write out.

Synthetic division is a shortcut that can be used when a polynomial is divided by a binomial.

It uses the coefficients in a tabular format, saving time.

The root of the binomial is used as a divisor, so a polynomial divided by, say, x − 3 will use a root of 3.

Synthetic division is best explained via example.

J. Garvin — Long and Synthetic Division of PolynomialsSlide 7/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Synthetic Division

Example Divide f (x) = x^3 − 2 x^2 + 4x − 1 by x − 2.

Since x − 2 is a factor, use a root of x = 2.

2 1 -2 4 -

  • 2 0 8 1 0 4 7

When x^3 − 2 x^2 + 4x − 1 is divided by x − 2, the quotient is x^2 + 4 and the remainder is 7. Therefore, f (x) = (x − 2)(x^2 + 4) + 7.

J. Garvin — Long and Synthetic Division of PolynomialsSlide 8/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Synthetic Division

Example

Divide g (x) = 2x^3 + x − 3 by x + 1.

Therefore, g (x) = (x + 1)(2x^2 − 2 x + 3) − 6.

J. Garvin — Long and Synthetic Division of Polynomials Slide 9/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Synthetic Division

Synthetic division works well when a polynomial is divided by a binomial of the form x − b. When the binomial has the form ax − b, the quotient will be off by a factor of a, and will need to be divided accordingly.

J. Garvin — Long and Synthetic Division of Polynomials Slide 10/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Synthetic Division

Example

Divide h(x) = 6x^3 + x^2 − 10 x + 5 by 3x − 1.

Since 3x − 1 = 0 has the root x = 13 , use this value.

1 3 6 1 -10^5

  • 2 1 - 6 3 -9 2

Dividing the quotient by 3, (6x^2 + 3x − 9) ÷ 3 = 2x^2 + x − 3.

Therefore, h(x) = (3x − 1)(2x^2 + x − 3) + 2.

J. Garvin — Long and Synthetic Division of PolynomialsSlide 11/

p o l y n o m i a l e q u a t i o n s & i n e q u a l i t i e s

Questions?

J. Garvin — Long and Synthetic Division of PolynomialsSlide 12/