Fundamentals of Mathematics
§3.2 - Adding, Subtracting, and Multiplying Polynomials
Ricky Ng
November 4, 2013
Ricky Ng
Fundamentals of Mathematics
Announcements
1
HW 15 is due tomorrow.
2
HW 16 is due on Thursday.
Ricky Ng
Fundamentals of Mathematics
Operations with Polynomials
Given two polynomials f (x) and g(x), we can perform the
following basic operations:
1
Addition / Subtraction
2
Multiplication
3
Division
After addition, subtraction, or multiplication, the resulting
function is still a polynomial; however, this is not true for
division in general. (Section 3.3)
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§3.2 - Adding, Subtracting, and Multiplying Po
Addition / Subtraction
There is only one rule to memorize when adding or subtraction
polynomials.
Rule
Group the like terms, then add or subtraction their coefficients
accordingly.
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§3.2 - Adding, Subtracting, and Multiplying Po
Examples
Perform the following operations and write the terms in
descending order (i.e. from highest to lowest degree).
(4x2 − 3x + 7) + (−x2 − 4)
(7x3 − 6x2 − 9) + (−2x2 + 4x + 8)
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§3.2 - Adding, Subtracting, and Multiplying Po
(6x − 4) − (5x2 + 4x − 9)
(4x3 − 5x + 2) − (4x3 − 2x2 − 4x + 1)
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§3.2 - Adding, Subtracting, and Multiplying Po
Popper 19: Question 1
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§3.2 - Adding, Subtracting, and Multiplying Po
Multiplication of Two Monomials
It is rather difficult to multiply two polynomials f (x) and g(x).
Let us first assume both f (x) and g(x) are monomials.
Rule (M1)
To multiply two monomials, we multiply them coefficient by
coefficient, and term by term.
For example,
(4x2 )(5x) = (4 × 5)(x2 × x) = 20x3 .
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§3.2 - Adding, Subtracting, and Multiplying Po
Examples
Multiply the following:
(3x2 )(−2x4 )
(−2x3 ) × (−6)
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§3.2 - Adding, Subtracting, and Multiplying Po
Multiplication with One Monomial
Now, suppose only f (x) is a monomial, and g(x) is a general
polynomial:
g(x) = b0 + b1 x + b2 x2 + · · · + bn xn .
Rule (M2)
To multiply them, we use the distributive property of g(x), then
multiply the monomials as in (M1).
For example,
(2x2 )(x + 4) = (2x2 )(x) + (2x2 )(4) = 2x3 + 8x2 .
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§3.2 - Adding, Subtracting, and Multiplying Po
Examples
Multiply the following:
(3x)(2x2 − x − 7)
(−2x4 )(−3x3 + 2x − 6)
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§3.2 - Adding, Subtracting, and Multiplying Po
Popper 19: Question 2
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§3.2 - Adding, Subtracting, and Multiplying Po
Popper 19: Question 3
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§3.2 - Adding, Subtracting, and Multiplying Po
Multiplication in General
In general, suppose both f (x) and g(x) are not monomials.
Rule (M3)
To multiply, we view f (x) as a sum of monomials, and
distribute each monomial over g(x) as in (M2).
For example,
(3x2 + 2)(2x2 + 3x) = (3x2 )(2x2 + 3x)+(2)(2x2 + 3x)
= (6x4 + 9x3 )+(4x2 + 6x)
= 6x4 + 9x3 + 4x2 + 6x.
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§3.2 - Adding, Subtracting, and Multiplying Po
Examples
Multiply (−2x − 1)(3x4 + 4x2 ).
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§3.2 - Adding, Subtracting, and Multiplying Po
Multiply (x2 + 2)2 .
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§3.2 - Adding, Subtracting, and Multiplying Po
Multiply (2x2 − x + 1)(x − 3).
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§3.2 - Adding, Subtracting, and Multiplying Po
Popper 19: Question 4
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§3.2 - Adding, Subtracting, and Multiplying Po
Popper 19: Question 5
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§3.2 - Adding, Subtracting, and Multiplying Po
Multiplication: Special Case with Binomials
When both f (x) and g(x) are binomials, using (M3) one can
get the following method known as FOIL. For example,
(2x + 1)(3x2 + x) = (2x)(3x2 ) + (2x)(x) + (1)(3x2 ) + (1)(x)
1
First: (2x)(3x2 )
2
Outer: (2x)(x)
3
Inner: (1)(3x2 )
4
Last: (1)(x)
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§3.2 - Adding, Subtracting, and Multiplying Po
Example
Use FOIL to multiply (x − 2)(3x + 1).
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§3.2 - Adding, Subtracting, and Multiplying Po
Popper 19: Question 6
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§3.2 - Adding, Subtracting, and Multiplying Po