Download Lesson 3

Lesson 3- Polynomials
Objectives :
- Definition
- Dividing Polynomials
Next Lesson
- Factor Theorem
- Remainder Theorem
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Polynomial
ax  bx
n
n 1
 cx
n2
 .................. jx  k
Real numbers called coefficients
n is the Degree of the polynomial
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Constant
Multiplying Polynomials
Expand all the terms
( x  2)( x  3x  6)  x  3x  6 x  2 x  6 x  12
2
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3
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2
2
Dividing Polynomials
This is trickier than multiplication
There are two main ways
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─
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Long Division
By Inspection
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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”
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Algebraic long division
Divide 2x³ + 3x² - x + 1 by x + 2
x  2 2x 3  3x 2  x  1
x + 2 is the
divisor
The quotient will
be here.
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2x³ + 3x² - x + 1
is the dividend
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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2x 2
x  2 2x 3  3x 2  x  1
Algebraic long division
Now multiply
2x²
by x + 2
and subtract
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2x 2
x  2 2x 3  3x 2  x  1
2x 3  4x 2
x 2
Algebraic long division
2x 2
x  2 2x 3  3x 2  x  1
Bring down the next
term, -x.
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2x 3  4x 2
x 2 x
Algebraic long division
Now divide –x², the
first term of –x² - x, by
x, the first term of the
divisor
2x 2 x
x  2 2x 3  3x 2  x  1
2x 3  4x 2
x 2 x
which gives –x.
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Algebraic long division
2x 2 x
x  2 2x 3  3x 2  x  1
2x 3  4x 2
Multiply –x by x + 2
and subtract
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x 2 x
x 2  2x
x
Algebraic long division
2x 2 x
x  2 2x 3  3x 2  x  1
2x 3  4x 2
Bring down the next term, 1
x 2 x
x 2  2x
x 1
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Algebraic long division
2x 2 x 1
x  2 2x 3  3x 2  x  1
Divide x, the first term of
x + 1, by x, the first term
of the divisor
2x 3  4x 2
x 2 x
x 2  2x
x 1
which gives 1
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Algebraic long division
2x 2 x 1
x  2 2x 3  3x 2  x  1
2x 3  4x 2
x 2 x
x 2  2x
Multiply x + 2 by 1
and subtract
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x 1
x 2
1
Algebraic long division
2x 2 x 1
x  2 2x 3  3x 2  x  1
The quotient is 2x² - x + 1
2x 3  4x 2
x 2 x
x 2  2x
The remainder is –1.
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x 1
x 2
1
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
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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
bx² + 4x² must be 3x²
so b must be -1.
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2 multiplied by 2x²
gives 4x²
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
bx² + 4x² must be 3x²
so b must be -1.
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2 multiplied by 2x²
gives 4x²
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
cx - 2x must be -x
so c must be 1.
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2 multiplied by -x
gives -2x
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
cx - 2x must be -x
so c must be 1.
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2 multiplied by -x
gives -2x
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
2 + d must be 1
so d must be -1.
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then add d
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
2 + d must be 1
so d must be -1.
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then add d
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.
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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
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Do the following
1.
(6 x 3  x 2  13x  7)  (2 x  1)
2.
(2 x 4  9 x 3  13x 2  17 x  15)  ( x  3)
3.
(3x 4  28x 3  43x 2  16 x  2)  ( x 2  7 x  4)
( x  3x  2)  ( x  1)
3
4.
2
Exercises C1/C2 Page 82 Ex 3A, Nos 3, 6, 9, 16 to 20
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