A polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables.

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1 A polynomial is an algebraic expression that has many terms connected by only the operations of +, -, and of variables. 2x x 7x +19 5x 2-7x + 19 x 2 1 x + 2 2x 3 y 4 z x + 2 2x The terms are the parts of the expression that are added or subtracted. Number of Terms or more Name monomial binomial trinomial no special name Example 3x 2 y 5 3x 2 + y x 2 + y 5 3x 5-2x 4 + 5x 3-6x 2 + 2x

2 The degree of a polynomial is the maximum number of variables that appear as factors in any one term. Constants do not vary Degree Name Example Memory Aid 0 constant 4 1 st linear 4x; 4x+4 A line has one dimension 2 nd quadratic 4x 2 ; 4x 2 +5x+2 A square is a quadrilateral 3 rd cubic 4x 3 ; 4xyz+2 A cube has 3 dimensions 4 th quartic 4x 4 ; 8x 2 y 2 +4x 3 A quarter is a fourth of a dollar 5 th quintic 4x 5 ; 3x+4tx 2 y 2 Quintuplets are 5 children 6 th or more no special name 4x 44

3 3x 2 3 is the numerical coefficient x is the base 2 is the exponent x 2 is the power Zero could have many different degrees, therefore it is considered to be a polynomial with no degree. 0 = 0x = 0x 2 = 0xy = 0x 92 = 0x 3 y 9 z 999

4 Multiplying binomials can be done by two different procedures. 1. FOIL 2. Double use of the distributive property (2x - 3)(3x + 1) FOIL mean First Outside Inside Last (2x - 3)(3x + 1) 6x 2 + 2x - 9x - 3

5 Multiplying binomials can be done by two different procedures. 1. FOIL 2. Double use of the distributive property (2x - 3)(3x + 1) To proceed through double use of the distributive property, think of (2x - 3) as one number,and distribute this number over the other binomial. (2x - 3) 3x + (2x - 3) 1 Then use the commutative property of multiplication to get 3x(2x - 3) + 1(2x - 3) Now distribute each of these new values 6x 2-9x + 2x - 9

6 In both cases, the polynomial can be simplified by combining like terms. 6x 2-7x - 9

7 Equations are two expressions connected by an equal sign. Solving an equation means writing its solution set. Solve: x 2 = 25 There are two answers, x = 5 and x = -5, so the solution is S = {-5, 5} Solve: 5x - 8 = 2x + 4 3x = 12 x = 4 This equation is equivalent to the original one. S = {4} They have the same solution set.

8 Before writing a solution set, you should check your answer by substituting the value(s) back into the original equation. An equation is not solved until you write the solution set. Adding (or subtracting) both members of an equation by the same number is justified by the Addition Property of Equality. Multiplying (or dividing) both members of an equation by the same number is justified by the Multiplication Property of Equality. While it is not necessary to justify each step when solving an equation, it will be necessary to do so when completing algebraic proofs - therefore learn the properties. No Work = No Credit, and the more work you show, the more partial credit you can earn.

9 Given the equation (x - 3)(x - 7) = 0: S = {3, 7} 1. Write the solution set. This equation contains a product which equals zero. The only way a product of two real numbers can equal zero is for one of the factors to equal zero. This fact is expressed as the converse of the Multiplication Property of Zero. x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 S = {3, 7}

10 Given the equation (x - 3)(x - 7) = 0: S = {3, 7} x - 7 = 0 S = {7} 2. Divide both members of the equation by (x - 3), and write the solution set for the transformed equation. (x - 3)(x - 7) = 0. (x - 3) (x - 3) x - 7 = 0 S = {7} 3. Is the transformed equation equivalent to the original equation? Explain. No, the solution sets are different.

11 Given the equation (x - 3)(x - 7) = 0: S = {3, 7} x - 7 = 0 S = {7} (x - 3)(x - 7)(x - 2) = 0 S = {2, 3, 7} 4. Multiply both members of the original equation by (x - 2), then solve it. (x - 3)(x - 7)(x - 2) = 0(x - 2) (x - 3)(x - 7)(x - 2) = 0 x - 3 = 0 or x - 7 = 0 or x - 2 = 0 x = 3 or x = 7 or x = 2 S = {2, 3, 7} 5. What number is a solution for the transformed equation that was not a solution of the original one? The new solution is 2.

12 Given the equation (x - 3)(x - 7) = 0: S = {3, 7} x - 7 = 0 S = {7} (x - 3)(x - 7)(x - 2) = 0 S = {2, 3, 7} 6. How can all three of these equations be equal to zero? Since x is a variable, there is the possibility that any of the factors could equal zero. When you multiply or divide by zero, you are performing an operation which seems perfectly correct, but which changes the solution set. Since it is the original equation you are trying to solve, you must be aware of these issues.

13 You should never divide an equation by something that can equal zero. DIVISION BY ZERO IS UNDEFINED and therefore NOT ALLOWED!!!!! When you divide by zero you can lose a valid solutions. When you multiply by zero, you are performing an irreversible step. This can lead to an extraneous solution. An extraneous solution is a solution which satisfies the transformed equation, but not the original equation. It is an extra solution. You will find which solutions are extraneous by checking your answer. You can have equations with no solutions. In this case the solution set is empty. S = Ø or S = { }

14 Solve: x - 2 = 3 Remember: absolute value is distance from zero. x - 2 = 3 means (x - 2) is three units form zero. This is three units to the left and three units to the right. x - 2 = 3 or x - 2 = -3 x = 5 or x = -1 S = {-1, 5}