2. CHAPTER R: Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
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3. R.3 Addition, Subtraction, and Multiplication of Polynomials
Identify the terms, coefficients, and the degree of a polynomial.
Add, subtract, and multiply polynomials.
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Polynomials
Polynomials are a type of algebraic expression.
Examples: y  6t
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5. Polynomials in One Variable
A polynomial in one variable is any expression of the type
where n is a nonnegative integer, an,…, a0 are real numbers, called coefficients. The parts of the polynomial separated by plus signs are called terms. The leading coefficient is an, and the constant term is a0. If an  0, the degree of the polynomial is n. The polynomial is said to be written in descending order, because the exponents decrease from left to right.
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Examples
Identify the terms of the polynomial.
4x7  3x5 + 2x2  9
The terms are: 4x7, 3x5, 2x2, and 9.
Find the degree of each polynomial.
Polynomial Degree
a) 7x5  3 5
b) x2 + 3x + 4x c)
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Degree of a Polynomial
An algebraic expression like 5a3b + 2ab – 1 is a polynomial in several variables. The degree of a term is the sum of the exponents of the variables in that term. The degree of a polynomial is the degree of the term of the highest degree.
The degrees of the terms of 5a3b + 2ab – 1 are 4, 2, and 0. The degree of the polynomial is 4.
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8. Addition and Subtraction
If two terms of an expression have the same variables raised to the same powers, they are called like terms, or similar terms.
Like Terms Unlike Terms
3y2 + 7y2 8c + 5b
4x3  2x3 9w  3y
We add or subtract polynomials by combining like terms.
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Examples
Add: (4x4 + 3x2  x) + (3x4  5x2 + 7)
= (4x4 + 3x4) + (3x2  5x2)  x + 7
= (4 + 3)x4 + (3  5)x2  x + 7
= x4  2x2  x + 7
Subtract: 8x3y2  5xy  (4x3y2 + 2xy)
= 8x3y2  5xy  4x3y2  2xy
= 4x3y2  7xy
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Multiplication
Multiplication is based on the distributive property.
Example
Multiply: (5x  1)(2x + 5)
= 5x(2x + 5) – 1(2x + 5)
= 10x2 + 25x  2x  5
= 10x2 + 23x  5
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Multiplication
To multiply two polynomials in general, we multiply each term of one by each term of the other and add the products.
Example: (3x3y  5x2y + 5y)(4y  6x2y)
3x3y(4y  6x2y)  5x2y(4y  6x2y) + 5y(4y  6x2y)
= 12x3y2  18x5y2  20x2y2 + 30x4y2 + 20y2  30x2y2
= 18x5y2 + 30x4y2 + 12x3y2  50x2y2 + 20y2
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Multiplication - FOIL
We can find the product of two binomials by multiplying the First terms, then the Outer terms, then the Inner terms, then the Last terms. Then we combine like terms if possible. This procedure is called FOIL.
Example: Multiply (x  5)(4x  1)
(x  5)(4x  1) = 4x2 – x – 20x + 5
= 4x2 – 21x + 5
F O I L
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Multiplication
Special Products of Binomials
(A + B)2 = A2 + 2AB + B2 Square of a sum
(A – B)2 = A2 – 2AB + B2 Square of a difference
(A + B)(A – B) = A2 – B2 Product of a sum and a difference
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14. Special Product Examples
Multiply: (6x  1)2
= (6x)2 – 2 • 6x + 12
= 36x2  12x + 1
Multiply: (2x  3)(2x + 3)
= (2x)2 – 32
= 4x2  9
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