Download Math 101 Lecture Notes Ch. 2.1 Page 1 of 4 2.1 Simplifying Algebraic

Math 101 Lecture Notes Ch. 2.1 2.1 Simplifying Algebraic Expressions Combining Like Terms An expression containing a number, a variable, or a product of numbers and variables is called a term. For example, –3, x, 5x, –3xy, –ab2c are terms. The number preceding the variable(s) in a term is called the coefficient. 5x –3xy x –ab2c has a coefficient of has a coefficient of has a coefficient of has a coefficient of 5 –3 1 –1 Like terms are terms that have the same variable factors raised to the same powers. For example, Like terms Unlike terms 2x, x 2x, y –8x2, 3x2 –8x2, 3x 5x2y, 3x2y 5x2y, 3xy2 5, 14 5, x We can use the distributive property to help us combine like terms. Recall that for any real numbers a, b, and c, we have a(b + c) = ab + ac and (b + c)a = ab + ac Suppose we want to simplify 2x + 7x. First, notice that by the distributive property (2 + 7)x = 2x + 7x By reversing the equation above and simplifying 2x + 7x = (2 + 7)x = 9x Example (a) Simplify –8x2 + 3x2 –8x2 + 3x2 = (–8 + 3)x2 = –5x2 Page 1 of 4 Math 101 Lecture Notes Ch. 2.1 Example (b) Simplify 3x – 9 + x – 2 – 5x 3x – 9 + x – 2 – 5x = 3x + (–9) + x + (–2) + (–5x) ß Rewrite as an equivalent sum of terms = 3x + x + (–5x) + (–9) + (–2) ß A pply the commutative property = (3 + 1 + (–5))x + (–11) ß Apply the distributive property = –1x + (–11) ß Simplify = –x – 11 ß Rewrite as an equivalent difference Demonstration Problems Simplify 1. (a) 3x + 5x 2. (a) –5xy – (–4xy) 1. (b) 2x + 12x 2. (b) –5ab – (–3ab) 3. (a) 2m + m 4. (a) –9x – 5 + 3 – 5x 3. (b) w + 4w 4. (b) –x – 8 – 7 + 9x 1
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Answers: 1. (b) 14x; 2. (b) –2ab; 3. (b) 5w; 4. (b) 8x – 15; 5. (b) (1/6)x Page 2 of 4 Math 101 Lecture Notes Ch. 2.1 Example (c) Simplify 3x – (5x + 5) 3x – (5x + 5) = 3x + –(5x + 5) ß Rewrite as an equivalent sum of terms = 3x + –1•(5x + 5) ß Insert “1” before the (5x + 5) = 3x + –1•(5x) + (–1)•5 ß Apply the distributive property = 3x + (–5x) + (–5) ß Simplify the products = –2x + (–5) ß Combine like terms = –2x – 5 ß Rewrite as an equivalent difference Demonstration Problems Simplify 6. (a) 10x – (2x – 5) 7. (a) (2x – 7) – (3x – 5) 8. (a) –7 – 4(3x – 2) Practice Problems Simplify 6. (b) 5x – (12x – 7) 7. (b) (3m – 7) – (2m – 3) 8. (b) –2 – 5(5x – 6) Answers: 6. (b) –7x + 7; 7. (b) m – 4; 8. (b) –25x + 28 Page 3 of 4 Math 101 Lecture Notes Ch. 2.1 Example (d) Translate to an algebraic expression and simplify. Four times the sum of a number and one. 4(x + 1) = 4x + 4 Example (e) Write a simplified expression for the perimeter of the figure. Perimeter = 2 • length + 2 • width 14 x + 3 Perimeter = 2 • (x + 3) + 2 • 14 = 2x + 6 + 28 = 2x + 34 Demonstration Problems Practice Problems Translate to an algebraic expression and Translate to an algebraic expression and simplify simplify 9. (a) The sum of twice a number and 5 Write a simplified expression for the perimeter 10. (a) 3x 2x – 5 9. (b) The sum of a number and –3 Write a simplified expression for the perimeter 10. (b) 1 – x 7 Answers: 9. (b) x – 3; 3x – 3; 10. (b) –2x + 16 Page 4 of 4