Algebraic Expressions An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols. Here are some examples of algebraic expressions. 5x²+2x-3, x+2, 1/3 xy-5y, 7(x-2)
Consider the example:5x²+x-7 The terms of the expression are separated by addition. There are 3 terms in this example and they are 5x², x and -7. The coefficient of a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1. The last term, -7, is called a constant since there is no variable in the term.
Distributive Property a ( b + c ) = ba + ca To simplify some expressions we may need to use the Distributive Property Do you remember it? Distributive Property
Practice Problem Try the Distributive Property on -7 ( x – 2 ). Be sure to multiply each term by –7. -7 ( x – 2 ) = x(-7) – 2(-7) = -7x + 14 Notice when a negative is distributed all the signs of the terms in the ( )’s change.
Examples with 1 and –1. Example 3: (x – 2) = 1( x – 2 ) = x(1) – 2(1) = x - 2 Notice multiplying by a 1 does nothing to the expression in the ( )’s. Example 4: -(4x – 3) = -1(4x – 3) = 4x(-1) – 3(-1) = -4x + 3 Notice that multiplying by a –1 changes the signs of each term in the ( )’s.
Like Terms Like terms are terms with the same variables raised to the same power. Hint: The idea is that the variable part of the terms must be identical for them to be like terms.
Examples Like Terms 5x, -14x -6.7xy, 02xy The variable factors are identical. Unlike Terms 5x, 8y The variable factors are not identical.
Combining Like Terms Recall the Distributive Property a (b + c) = b(a) +c(a) To see how like terms are combined use the Distributive Property in reverse. 5x + 7x = x (5 + 7) = x (12) = 12x
Example All that work is not necessary every time. Simply identify the like terms and add their coefficients. 4x + 7y – x + 5y = 4x – x + 7y +5y = 3x + 12y
Both Skills This example requires both the Distributive Property and combining like terms. 5(x – 2) –3(2x – 7) Distribute the 5 and the –3. x(5) - 2(5) + 2x(-3) - 7(-3) 5x – 10 – 6x + 21 Combine like terms. - x+11
Evaluating Expressions Remember to use correct order of operations. Evaluate the expression 2x – 3xy +4y when x = 3 and y = -5. To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number.
Example Evaluate 2x–3xy +4y when x = 3 and y = -5. Substitute in the numbers. =2(3) – 3(3)(-5) + 4(-5) Use correct order of operations. =6 + 45 – 20 =51 – 20 =31
Evaluating Example Remember correct order of operations. Substitute in the numbers.
Factoring Algebraic Expression To factor a number means to rewrite it as the product of smaller numbers. To factor an algebraic expression means to rewrite it as the product of simpler algebraic expressions. We factor algebraic expressions to simplify the expressions and to help solve equations. The number 36 can be factored several different ways
Factoring Algebraic Expressions To factor an algebraic expression we start by looking for common factors in each of its terms. If there are factors common to each term we can factor them out of each term. Here each term has 2 as a common factor and also x as a common factor. When we factor 2x out of each term we get This expression cannot be factored any further. To be sure that you have factored correctly do the multiplication 2x(3x-1) and see that you get the original expression back again.
Factoring Algebraic Expressions Even when there is no factor common to each term of an algebraic expression we can often still factor it into two or more simpler algebraic expressions. For example: can be factored into This occurs frequently when we want to factor quadratic expressions. A quadratic expression is one of the form
Factoring Algebraic Expressions To factor the expression start by looking at the factor pairs of 12. We are looking for a pair of factors which add up to equal 8. Because 12 is positive we are looking for factor pairs which are either both positive or both negative; because we want them to add up to a positive 8 we only need to look at the positive factor pairs of 12. The positive factor pairs of 12 are: 12 and 1 6 and 2 4 and 3 Since 6 + 2 = 8 this is the pair we are looking for and we can factor the original expression into:
Factoring by Grouping Now we look at how to factor algebraic expressions that have more than three terms. To do this we use a technique called factoring by grouping. We will group the terms into two (or more) groups and factor each group separately. We hope that this results in a factor common to each group which can then be factored out of each group. Factor Group the first two terms together and the second two terms together and factor each group. Group and factor Group and factor
Factoring Quadratics when a≠1 Start by multiplying the leading coefficient (6) to the constant (-20) (6)(-20) = -120 Now look at factor pairs of -120 which add up to -7. One factor must be positive and the other must be negative. Also, the larger factor must be the negative one and the smaller factor must be the positive one
Factoring Quadratics when a≠1 The factor pairs of -120 which meet the requirements are -120 and 1-60 and 2 -40 and 3-30 and 4 -24 and 5-20 and 6 -15 and 8-12 and 10 Because -15 + 8 = -7 this is the pair that we are looking for and we can rewrite Now use factoring by grouping