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MHF4U: Advanced Functions
Long and Synthetic Division of Polynomials
J. Garvin
Slide 1/
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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/
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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/
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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/
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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/
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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/
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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/
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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 -
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/
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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/
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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/
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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
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/
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Questions?
J. Garvin — Long and Synthetic Division of PolynomialsSlide 12/
