Module :MA0001NPFoundation Mathematics

Lecture Week 3

Significant FiguresandRounding

Figures

Algebraic Expression, Simplifying Algebraic

Expressions & Factorizing

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-7The 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

Examples

Example 1: 6(x + 2)Distribute the 6.

6 (x + 2) = x(6) + 2(6) = 6x + 12

Example 2: -4(x – 3)Distribute the –4.

-4 (x – 3) = x(-4) –3(-4) = -4x + 12

Practice ProblemTry 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.

ExamplesLike Terms5x , -14x

-6.7xy , 02xy

The variable factors areidentical.

Unlike Terms5x , 8y

The variable factors are not identical.

22 8,3 xyyx

Combining Like TermsRecall 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

Collecting Like Terms Example

31316

terms.likeCombine

31334124

terms.theReorder

33124134

2

22

22

yxx

yxxxx

xxxyx

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

Simplifying Example 431062

1 xx

12353

3432

110

2

16

xx

xx

76 x

Evaluating Expressions

Remember to use correct order of operations.

Evaluate the expression 2x – 3xy +4y whenx = 3 and y = -5.

To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number.

ExampleEvaluate 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.

1and2when34Evaluate 22 yxyxyx

22 131242

Substitute in the numbers.

131244

384

15

Common MistakesIncorrect Correct

Factoring Algebraic Expression

9436

12336

18236

36136

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

22 32332236

33436

92236

6636

Factoring Algebraic Expressions

To factor an algebraic expression we start by looking for commonfactors in each of its terms. If there are factors common to each termwe 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

xx 26 2

13226 2 xxxx

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:

1522 xxcan be factored into 3 5x x

This occurs frequently when we want to factor quadratic expressions.

A quadratic expression is one of the form cbxax 2

Factoring Algebraic Expressions 1282 xxTo 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 3Since 6 + 2 = 8 this is the pair we are looking for and we can factor the original expression into:

26 xx

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. Factorx3 7x2 3x 21

Group the first two terms together and the second two terms together and factor each group.

2137 23 xxxGroup andfactor

Group andfactor

x2 x 7 3 x 7

Practice Factor 6x3 27x2 10x 45

Factoring Quadratics when a≠1

6x2 7x 20

Start by multiplying the leading coefficient (6) to the constant (-20)(6)(-20) = -120Now 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≠1The 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 canrewrite

6x2 7x 20 6x2 15x 8x 20

Now use factoring by grouping

4352

524523

208156 2

xx

xxx

xxx

Practice Factor

35 + (-3) = 32

213225 xx

Practice

Factorize

Do any of those pairs add up to 22?

Rewrite and factor!

3512

9124

31110

152

12228

2

2

2

2

2

xx

xx

xx

yy

xx Look for f

Practice

Factorize

Do any of those pairs add up to 22?

Rewrite and factor!

90

)()1(

15143

12

107

2

222

22

2

2

xx

xbaxab

mmxx

xx

xx Look for f