### Introduction to Polynomials and Polynomial Functions

```Sect. 5.1
Polynomials and Polynomial Functions
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Definitions
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Polynomial Functions
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Terms
Degree of terms and polynomials
Evaluating
Graphing
Simplifying by Combining Like Terms
5.1
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Definitions
An algebraic term is a number or a product of a number and a
variable (or variables) raised to a positive power.
Examples: 7x or -11xy2 or 192 or z
A constant term contains only a number
A variable term contains at least one variable
and has a numeric part and a variable part
A polynomial expression is:
one or more terms separated by addition or subtraction;
any exponents must be whole numbers;
no variable in any denominator.
Monomial – one term: -47 or 7x or 92xyz5
Binomial – two terms: a + b or 7x5 – 44
Trinomial – three terms: x2 + 6x + 9 or a + b – 55c
5.1
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Ordering a Polynomial’s Terms
If there are multiple variables, one must be specified
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Descending Order – Variable with largest
exponent leads off (this is the Standard)
Ascending Order – Constant term leads off
Arrange in ascending order:
Arrange in descending order of x:
12  2 x  7 x  x
2
y  2  5 x  3x y  7 xy
12  7 x  x  2 x
3
3x y  5 x  7 xy  y  2
3
2
4
2
3
5.1
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3
2
4
3
2
DEGREE
 Of a Monomial: (a Single Term)
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Several variables – the degree is the sum of their exponents
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One variable – its degree is the variable’s exponent
Non-zero constant – the degree is 0
The constant 0 has an undefined degree
Of a Polynomial: (2 or more terms added or subtracted)
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Its degree is the same as the degree of the term in the
polynomial with largest degree (leading term?).
5.1
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Polynomial Functions
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Equations in one variable:
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f(x) = 2x – 3
(straight line)
g(x) = x2 – 5x – 6 (parabola)
h(x) = 3x3 + 4x2 – 2x + 5
Evaluate by substitution:
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f(5) = 2(5) – 3 = 10 – 3 = 7
g(-2) = (-2)2 – 5(-2) – 6 = 4 + 10 – 6 = 8
h(-1) = 3(-1)3 + 4(-1)2 – 2(-1) + 5 = -3+ 4+2+5= 8
5.1
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Opposites of Monomials
The opposite of a monomial has a different sign
The opposite of 36 is -36
The opposite of -4x2 is 4x2
Monomial:Opposite:
-2
5y
¾y5
-x3
0
2
-5y
-¾y5
x3
0
5.1
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Writing Any Polynomial as a Sum
-5x2 – x is the same as -5x2 + (-x)
Keep the negative sign with the monomial
 4x5 – 2x6 – 4x + 7 is
 4x5 + (-2x6) + (-4x) + 7
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You try it:
 -y4 + 3y3 – 11y2 – 129
 -y4 + 3y3 + (-11y2) + (-129)
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5.1
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Identifying Like Terms
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When several terms in a polynomial have the
same variable(s) raised to the same power(s),
we call them like terms.
3x + y – x – 4y + 6x2 – 2x + 11xy
Like terms: 3x, -x, -2x
Also: y, -4y
You try: 6x2 – 2x2 – 3 + x2 – 11
Like terms: 6x2, -2x2, x2
Also: -3, -11
5.1
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Collecting Like Terms (simplifying)
The numeric factor in a term is its coefficient.
 3x + y – x – 4y + 6x2 – 2x
3
1 -1 -4 6 -2
 You can simplify a polynomial by collecting
like terms, summing their coefficients
 Let’s try: 6x2 – 2x2 – 3 + x2 – 11
 Sum of: 6x2 + -2x2 + x2 is 5x2
 Sum of: -3 + -11 is -14
 Simplified polynomial is: 5x2 – 14
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5.1
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Collection Practice
2x3 – 6x3 = -4x3
 5x2 + 7 + 4x4 + 2x2 – 11 – 2x4 = 2x4 + 7x2 – 4
 4x3 – 4x3 = 0
 5y2 – 8y5 + 8y5 = 5y2
 ¾x3 + 4x2 – x3 + 7 = -¼x3 + 4x2 + 7
 -3p7 – 5p7 – p7 = -9p7
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5.1
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Missing Terms
x3 – 5 is missing terms of x2 and x
 So what!
when you start adding & subtracting polynomials
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Write the expression above in either of two ways:
 With 0 coefficients: x3 + 0x2 + 0x – 5
 With space left:
x3
–5
5.1
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remove parentheses and combine like terms.
 (2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 = 9x – 3
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(5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2 – x – 6
= 5x2 – 4x – 4
 This is called the horizontal method because
you work left to right on the same “line”
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5.1
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To add polynomials vertically, remove parentheses,
put one over the other lining up like terms, add terms.
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(2x – 5) + (7x + 2) = 2x – 5
+ 7x + 2
Add the matching columns 9x – 3
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(5x2 – 3x + 2) + (-x – 6) =
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This is called the vertical method because you work from top to
5.1
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bottom. More than 2 polynomials
can be added at the same time.
5x2 – 3x + 2
+
–x–6
5x2 – 4x – 4
Opposites of Polynomials
The opposite of a polynomial has a reversed sign for
each monomial
The opposite of
The opposite of
Polynomial:
-x + 2
3z – 5y
¾y5 + y5 – ¼y5
-(x3 – 5)
y + 36 is -y – 36
-4x2 + 2x – 4 is 4x2 – 2x + 4
Opposite:
x–2
-3z + 5y
-¾y5 – y5 + ¼y5
x3 – 5
5.1
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Subtracting Polynomials
To subtract polynomials, add the opposite of the
second polynomial.
 (7x3 + 2x + 4) – (5x3 – 4) add the opposite!
(7x3 + 2x + 4) + (-5x3 + 4)
 Use either horizontal or vertical addition.
 Sometimes the problem is posed as subtraction:
x2 + 5x +6 make it addition x2 + 5x +6
- (x2 + 2x) _ of the opposite
-x2 – 2x__
3x +6
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5.1
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Subtractions for You
(5 x 2  4)  (2 x 2  3x  1)
 5x 2
4
  2 x 2  3x  1
 7 x 2  3x  5
(4 x 2 y  6 x 3 y 2  x 2 y 2 )  (4 x 2 y  x 3y 2  3x 2 y 3  8 x 2 y 2 )
4 x 2 y  6 x3 y 2
 x2 y 2
  4 x 2 y  x 3y 2  3x 2 y 3  8 x 2 y 2
 7 x 3y 2  3x 2 y 3  9 x 2 y 2
5.1
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What Next?
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Section 5.2 –
Multiplication of Polynomials
5.1
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