1. Sect. 5.1 Polynomials and Polynomial Functions
Definitions
Terms
Degree of terms and polynomials
Polynomial Functions
Evaluating
Graphing
Simplifying by Combining Like Terms
Adding & Subtracting Polynomials
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2. 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 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: or 7x or 92xyz5
Binomial – two terms: a + b or 7x5 – 44
Trinomial – three terms: x2 + 6x or a + b – 55c
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3. Ordering a Polynomial’s Terms If there are multiple variables, one must be specified
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:
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4. Of a Monomial: (a Single Term)
DEGREE
Of a Monomial: (a Single Term)
Several variables – the degree is the sum of their exponents
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)
Its degree is the same as the degree of the term in the polynomial with largest degree (leading term?).
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5. Polynomial Functions Equations in one variable:
f(x) = 2x – (straight line)
g(x) = x2 – 5x – (parabola)
h(x) = 3x3 + 4x2 – 2x + 5
Evaluate by substitution:
f(5) = 2(5) – 3 = 10 – 3 = 7
g(-2) = (-2)2 – 5(-2) – 6 = – 6 = 8
h(-1) = 3(-1)3 + 4(-1)2 – 2(-1) + 5 = = 8
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6. 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 2
5y -5y
¾y5 -¾y5
-x3 x3
0 0
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7. Writing Any Polynomial as a Sum
-5x2 – x is the same as -5x2 + (-x)
Replace subtraction with addition: Keep the negative sign with the monomial
4x5 – 2x6 – 4x is
4x5 + (-2x6) + (-4x) + 7
You try it:
-y4 + 3y3 – 11y2 – 129
-y4 + 3y3 + (-11y2) + (-129)
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8. Identifying Like Terms
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
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9. Collecting Like Terms (simplifying)
The numeric factor in a term is its coefficient.
3x + y – x – 4y + 6x2 – 2x
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: is -14
Simplified polynomial is: 5x2 – 14
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10. Collection Practice 2x3 – 6x3 = -4x3
5x x4 + 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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11. Missing Terms x3 – 5 is missing terms of x2 and x So what!
Leaving space for missing terms will help you when you start adding & subtracting polynomials
Write the expression above in either of two ways:
With 0 coefficients: x3 + 0x2 + 0x – 5
With space left: x – 5
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12. Adding 2 Polynomials - Horizontal
To add polynomials, remove parentheses and combine like terms.
(2x – 5) + (7x + 2) = 2x – 5 + 7x + 2 = 9x – 3
(5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x + 2 – x – = 5x2 – 4x – 4
This is called the horizontal method because you work left to right on the same “line”
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13. Adding 2 Polynomials - Vertical
To add polynomials vertically, remove parentheses, put one over the other lining up like terms, add terms.
(2x – 5) + (7x + 2) = 2x – x Add the matching columns 9x – 3
(5x2 – 3x + 2) + (-x – 6) = 5x2 – 3x – x – x2 – 4x – 4
This is called the vertical method because you work from top to bottom. More than 2 polynomials can be added at the same time.
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14. Opposites of Polynomials
The opposite of a polynomial has a reversed sign for each monomial
The opposite of y is -y – 36
The opposite of -4x2 + 2x – 4 is 4x2 – 2x + 4
Polynomial: Opposite:
-x + 2 x – 2
3z – 5y -3z + 5y
¾y5 + y5 – ¼y5 -¾y5 – y5 + ¼y5
-(x3 – 5) x3 – 5
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15. 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 make it addition x2 + 5x (x2 + 2x) _ of the opposite x2 – 2x__ x +6
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16. Subtractions for You
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17. What Next?
Section 5.2 – Multiplication of Polynomials
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