Dividing polynomials

1
Dividing polynomials
This PowerPoint presentation demonstrates two
different methods of polynomial division.
Click here to see algebraic long division
Click here to see dividing in your head
2
Algebraic long division
Divide 2x³ 3x² - x 1 by x 2
x 2 is the divisor
2x³ 3x² - x 1 is the dividend
The quotient will be here.
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Algebraic long division
First divide the first term of the dividend, 2x³,
by x (the first term of the divisor).
This gives 2x². This will be the first term of
the quotient.
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Algebraic long division
Now multiply 2x² by x 2
and subtract
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Algebraic long division
Bring down the next term, -x.
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Algebraic long division
Now divide x², the first term of x² - x, by x,
the first term of the divisor
which gives x.
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Algebraic long division
Multiply x by x 2
and subtract
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Algebraic long division
Bring down the next term, 1
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Algebraic long division
Divide x, the first term of x 1, by x, the
first term of the divisor
which gives 1
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Algebraic long division
Multiply x 2 by 1
and subtract
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Algebraic long division
The quotient is 2x² - x 1
The remainder is 1.
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Dividing polynomials
Click here to see this example of algebraic long
division again
Click here to see dividing in your head
Click here to end the presentation
13
Dividing in your head
Divide 2x³ 3x² - x 1 by x 2
When a cubic is divided by a linear expression,
the quotient is a quadratic and the remainder, if
any, is a constant.
Let the quotient by ax² bx c
Let the remainder be d.
2x³ 3x² - x 1 (x 2)(ax² bx c) d
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Dividing in your head
The first terms in each bracket give the term in
x³
2x³ 3x² - x 1 (x 2)(ax² bx c) d
x multiplied by ax² gives ax³
so a must be 2.
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Dividing in your head
The first terms in each bracket give the term in
x³
2x³ 3x² - x 1 (x 2)(2x² bx c) d
x multiplied by ax² gives ax³
so a must be 2.
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Dividing in your head
Now look for pairs of terms that multiply to give
terms in x²
2x³ 3x² - x 1 (x 2)(2x² bx c) d
x multiplied by bx gives bx²
2 multiplied by 2x² gives 4x²
bx² 4x² must be 3x²
so b must be -1.
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Dividing in your head
Now look for pairs of terms that multiply to give
terms in x²
2x³ 3x² - x 1 (x 2)(2x² -1x c) d
x multiplied by bx gives bx²
2 multiplied by 2x² gives 4x²
bx² 4x² must be 3x²
so b must be -1.
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Dividing in your head
Now look for pairs of terms that multiply to give
terms in x
2x³ 3x² - x 1 (x 2)(2x² - x c) d
x multiplied by c gives cx
2 multiplied by -x gives -2x
cx - 2x must be -x
so c must be 1.
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Dividing in your head
Now look for pairs of terms that multiply to give
terms in x
2x³ 3x² - x 1 (x 2)(2x² - x 1) d
x multiplied by c gives cx
2 multiplied by -x gives -2x
cx - 2x must be -x
so c must be 1.
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Dividing in your head
Now look at the constant term
2x³ 3x² - x 1 (x 2)(2x² - x 1) d
2 multiplied by 1 gives 2
then add d
2 d must be 1
so d must be -1.
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Dividing in your head
Now look at the constant term
2x³ 3x² - x 1 (x 2)(2x² - x 1) - 1
2 multiplied by 1 gives 2
then add d
2 d must be 1
so d must be -1.
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Dividing in your head
2x³ 3x² - x 1 (x 2)(2x² - x 1) - 1
The quotient is 2x² - x 1 and the remainder is
1.
23
Dividing polynomials
Click here to see this example of dividing in
your head again
Click here to see algebraic long division
Click here to end the presentation