Algebraic Expressions & Polynomials
Chapter 5 Sections 5.1-5.3
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Fundamental Operations5.1
• In arithmetic, we perform mathematical operations with specific
numbers.
• In algebra, we perform these same basic mathematical operations
with numbers and variables—letters that represent unknown
quantities.
• To begin our study of algebra, some basic mathematical principles
that you will apply are listed below.
• Note that “≠” means “is not equal to.”
Fundamental Operations
• Basic Mathematical Principles
• 1. a + b = b + a (Commutative Property for Addition)
• 2. ab = ba (Commutative Property for Multiplication)
• 3. (a + b) + c = a + (b + c) (Associative Property for Addition)
• 4. (ab)c = a(bc) (Associative Property for Multiplication)
• 5. a(b + c) = ab + ac, or (b + c)a = ba + ca (Distributive Property)
Fundamental Operations
• 6. a + 0 = a
• 7. a 0 = 0
• 8. a + (–a) = 0 (Additive Inverse)
• 9. a 1 = a
• 10. a = 1 (a ≠ 0) (Multiplicative Inverse)
Fundamental Operations
• In mathematics, letters are often used to represent numbers.
• Thus, it is necessary to know how to indicate arithmetic operations
and carry them out using letters.
• Addition: x + y means add x and y.
• Subtraction: x – y means subtract y from x or add the
negative of y to x; that is, x + (–y).
• Multiplication: xy or x y or (x)(y) or (x)y or x(y) means
multiply x by y.
Fundamental Operations
• Division: x y or means divide x by y, or find a
number z such that zy = x.
• Exponents: xxxx means use x as a factor 4 times, which is
abbreviated by writing x4.
• In the expression x4, x is called the base, and
4 is called the exponent.
• For example, 24 means 2 2 2 2 = 16.
Fundamental Operations
• Order of Operations
• 1. Perform all operations inside parentheses first. If the problem contains a fraction bar, treat the numerator and the denominator separately.
• 2. Evaluate all powers, if any. For example, 6 23 = 6 8 = 48.
• 3. Perform any multiplications or divisions in order, from left to right.
• 4. Do any additions or subtractions in order, from left to right.
Fundamental Operations
• Evaluate: 4 – 9(6 + 3) (–3).
• = 4 – 9(9) (–3)
• = 4 – 81 (–3)
• = 4 – (–27)
• = 31
Example 1
Add within parentheses.
Multiply.
Divide.
Subtract.
• To evaluate an expression, replace the letters
with given numbers; then do the arithmetic using
the order of operations.
• The result is the value of the expression.
• Evaluate ab/3c + c if a=6 b=10 c= -5
• 6(10)/3(-5) + (-5)= 60/-15 + (-5)= -4 + (-5)=
• -9
Fundamental Operations
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Simplifying Algebraic Expressions5.2
• Parentheses are often used to clarify the order of
operations when the order of operations is complicated
or may be ambiguous.
• Sometimes it is easier to simplify such an expression by
first removing the parentheses—before doing the
indicated operations.
Simplifying Algebraic Expressions
• Two rules for removing parentheses are as follows
• Removing Parentheses
• 1. Parentheses preceded by a plus sign may be removed without changing the signs of the terms within. Think of using the Distributive Property, a(b + c) = ab + ac, and
multiplying each term inside the parentheses by 1. That is,
• 3w + (4x + y) = 3w + 4x + y
Simplifying Algebraic Expressions
• 2. Parentheses preceded by a minus sign may be removed
if the signs of all the terms within the parentheses are
changed; then the minus sign that preceded the
parentheses is dropped. Think of using the Distributive
Property, a(b + c) = ab + ac, and multiplying each term
inside the parentheses by –1. That is,
• 3w – (4x – y) = 3w – 4x + y
• (Notice that the sign of the term 4x inside the
parentheses is not written. It is therefore understood to
be plus.)
Simplifying Algebraic Expressions
• Remove the parentheses from the expression
• 5x – (– 3y + 2z).
• 5x – (– 3y + 2z) = 5x + 3y – 2z
Example 1
Change the signs of all of the terms within parentheses; then drop the minus sign that precedes the parentheses.
• A term is a single number or a product of a number and one or more letters raised to powers. The following are examples of terms:
• 5x, 8x2, – 4y, 15, 3a2b3, t
• The numerical coefficient is the numerical factor of a term.
• The numerical factor of the term 16x2 is 16.• The numerical coefficient of the term – 6a2b is – 6.• The numerical coefficient of y is 1.
Simplifying Algebraic Expressions
• Terms are parts of an algebraic expression separated by
plus and minus signs.
• For example, 3xy + 2y + 8x2 is an expression consisting
of three terms.
Simplifying Algebraic Expressions
Like Terms
• Terms with the same variables with exactly the same exponents are called like terms.
• For example, 4x and 11x have the same variables and are like terms.
• The terms – 5x2y3 and 8x2y3 have the same variables with the same exponents and are like terms.
• The terms 8m and 5n have different variables, and the terms 7x2 and 4x3 have different exponents, so these are unlike terms.
Like Terms
• The following table gives examples of like terms and unlike terms.
Example 3
• Like terms that occur in a single
expression can be combined into one term
by combining coefficients (using the
Distributive Property).
• Thus,
• ba + ca = (b + c)a.
Like Terms
• Combine the like terms 2x + 3x.
• 2x + 3x = (2 + 3)x
• = 5x
Example 4
• Some expressions contain parentheses that must be removed before combining like terms. Follow the order of operations.
•
• a(b + c) = ab + ac.
• The Distributive Property is applied to remove
parentheses when a number, a letter, or some product
precedes the parentheses.
Like Terms
• Simplify: 3x + 5(x – 3).
• 3x + 5(x – 3) = 3x + 5x – 15
• = 8x – 15
Example 12Apply the Distributive Property by multiplying each term within the parentheses by 5.
Combine like terms.
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Addition and Subtraction of Polynomials5.3
• A monomial, or term, is any algebraic expression that contains only
products of numbers and variables, which have nonnegative integer
exponents.
• The following expressions are examples of monomials:
• 2x, 5, –3b,
• A polynomial is either a monomial or the sum or difference of
unlike monomials. We consider two special types of polynomials.
Addition and Subtraction of Polynomials
• A binomial is a polynomial that is the sum or difference of two
unlike monomials. A trinomial is the sum or difference of three
unlike monomials.
• The following table shows examples of monomials, binomials, and
trinomials.
Addition and Subtraction of Polynomials
• Expressions that contain variables in the denominator are not polynomials.
• For example,
• and
• are not polynomials.
Addition and Subtraction of Polynomials
• Find the degree of each monomial:
• a. –7m, b. 6x2, c. 5y3, d. 5.
• a. –7m has degree 1.
• b. 6x2 has degree 2.
• c. 5y3 has degree 3.
• d. 5 has degree 0
Example 1
The exponent of m is 1.
The exponent of x is 2.
The exponent of y is 3.
5 may be written as 5x0.
• A polynomial is in decreasing order if each term is of some degree less than the preceding term.
• The following polynomial is written in decreasing order:
• 4x5 – 3x4 – 4x2 – x + 5
Addition and Subtraction of Polynomials
exponents decrease
• A polynomial is in increasing order if each term is of some degree larger than the preceding term.
• The following polynomial is written in increasing order:
• 5 – x – 4x2 – 3x4 + 4x5
• Adding Polynomials• To add polynomials, add their like terms.
Addition and Subtraction of Polynomials
exponents increase
• Add: (3x + 4) + (5x – 7).
• (3x + 4) + (5x – 7) = (3x + 5x) + [4 + (–7)]
• = 8x – 3
Example 3
Add the like terms.
• Subtracting Polynomials
• To subtract two polynomials, change all
the signs of the terms of the second
polynomial and then add the two resulting
polynomials.
Addition and Subtraction of Polynomials
• Subtract: (5a – 9b) – (2a – 4b).
• (5a – 9b) – (2a – 4b)
• = (5a – 9b) + (–2a + 4b)
• = [5a + (–2a)] + [(–9b) + 4b]
• = 3a – 5b
Example 7
Add the like terms.
Change all the signs ofthe terms of the secondpolynomial and add.
Group Practice Problems
•Page 241
•3, 4, 5, 7