MATH 010 JIM DAWSON
1.1 INTRODUCTION TOINTEGERS • This section is an introduction to: • Positive Integers • Negative Integers • Opposites • Additive Inverse • Absolute Value
1.2 ADDING AND SUBTRACTING INTEGERS • If the signs are the same : ADD the absolute values and place the common sign in the answer. • 6+7=13 • - 13+(-5)= -18
If the signs are different : SUBTRACT the absolute value of the smaller number from the absolute value of the larger number. Place the sign of the larger number using absolute value in the answer. 14+(-6)= 8 -21+10= -11
1.3 MULTIPICATION AND DIVISION OF INTEGERS • Determine the sign of the answer first: Count the negative signs: even number of negative signs – the answer is positive An odd number of negative signs- the answer is negative
2.Multiply or Divide using the absolute values -5 x (-6)=30 7 x (-4)= -28
1.4 REVIEW OF FRACTIONS AND DECIMALS WITH SIGNS • The rules for sign are the same for fractions and decimals as they were for integers.
CONVERTING BETWEEN FRACTIONS, DECIMALS, AND PERCENTS • Change a percent to a decimal. • Move the decimal point TWO places from right to left. • Change a decimal to percent. • Move the decimal point TWO places from left to right.
FRACTION TO A PERCENT • Change the fraction to a decimal(numerator divided by denominator) and move the decimal point TWO places from left to right.
CHANGE A PERCENT WITH A FRACTION TO A FRACTION Drop the % and multiply by 1 over 100.
EXPONENTIAL NOTATION AND SOLVING EXPONENTS • If the base is negative and does not have parentheses around it the sign of the answer is ALWAYS negative. • If the base is negative and has parentheses around it; look at the exponent to find the sign of the answer • Even numbered exponent: positive answer • Odd numbered exponent: negative answer
ORDER OF OPERATIONS AGREEMENT • Priority #1-GROUPING SYMBOLS • Priority #2- EXPONENTS • Priority #3- MULTIPLY AND DIVIDE AS THEY OCCUR FROM LEFT TO RIGHT • Priority #4- ADD AND SUBTRACT AS THEY OCCUR FROM LEFT TO RIGHT
TRANSLATE AND SIMPLIFY • The translation must be done first and then simplify using the rules learned previously in the chapter. • The answer must be in descending order.
2.1 EVALUATING VARIABLE EXPRESSIONS • COMBINING LIKE TERMS • Add or subtract the terms with the same variable part • Place the answer in descending order • 2a+3b-4a+7b=2a-4a+3b+7b • -2a+10b
2.2 SIMPLIFYING VARIABLE EXPRESSIONS • Combining like terms • Combine the terms with the same variable part or the constants • -3a+7+5a-9=-3a+5a and7-9 • 2a-2
MULTIPLYING VARIABLE TERMS • Multiply the number parts and bring the variable into the answer. • -3x(5)=-3(5)=-15x • 7(-4a)=-7(4)=-28a • (-2b)(-6)=-2(-6)=12b
MULTIPLYING VARIABLE TERMS • Multiply the number parts and bring the variable into the answer. • -6(-4a)=-6(-4)= 24a • (-5x)(-3)=(-5)(-3)=15x
APPLYING THE DISTRIBUTIVE PROPERTY • The Distributive Property is used to remove parentheses. • If the terms inside the parentheses are different, multiply the term on the outside by every term on the inside. • Place the answer in descending order. • 3(2x-4)=3(2x) and 3(-4)=6x-12
If you cannot combine like terms inside the parentheses, multiply the outside term by each term inside the parentheses. • -3(4x+2)=-3(4x) and –3(2) • -12x-6 • Place the answer in descending order
SIMPLIFYING A GENERAL VARIABLE EXPRESSION • Use the Distributive Property to remove parentheses and brackets • Combine like terms when possible • Place the answer in descending order • Be careful with sign!
2.3 TRANSLATING VERBAL EXPRESSIONS • Memorize the expressions on p.67 • Rules for Parentheses • Use parentheses to infer multiplication when needed • Use parentheses to separate two processes together not separated by a number or a variable
Use parentheses to separate a more than or less than phrase with a number and letter next to the phrase from any other phrase in the expression
EXAMPLES OF TRANSLATING VERBAL EXPRESSIONS • 7 ADDED TO 3 LESS THAN A NUMBER • 7+(n-3) • 4 TIMES THE DIFFERENCE BETWEEN A NUMBER AND 4 • 4(n-4) • THE SUM OF 2 AND THE PRODUCT OF A NUMBER AND 9
EXAMPLES OF TRANSLATING • 2+9x • 6 TIMES THE TOTAL OF A NUMBER AND 8 • 6(n+8) • 5 INCREASED BY THE DIFFERENCE BETWEEN 10 TIMES a AND THREE • 5+(10a-3)
TRANSLATE AND SIMPLIFY • Translate the verbal expression FIRST and then simplify using the rules that were applied earlier in the chapter. • The answer must be in descending order if the expression was able to be simplified.
DEFINING THE UNKNOWNS • In order to define an unknown quantity, assign a variable to that quantity, and then attempt to express other unknown quantities in terms of the same variable. • These are equations of one variable; therefore, the same variable must be used when defining any unknowns.
3.1 SOLVING EQUATIONS OF THE FORM x+a=b • The Addition Property of Equations • The goal is to solve for the unknown • VARIABLE = CONSTANT • Find the number that is on the same side of the equation as the variable and use the opposite process on both sides of the equation to solve the unknown quantity.
SOLVING EQUATIONS USING THE ADDITION PROPERTY x+a=b • X+4=12 • Find the number that is on the same side of the equation as x and use the opposite process to remove the number from the x side . The number to be removed is 4. It`s opposite is –4. The unknown can be solved by the following; x+4-4=12-4;x=8 • You must do the same thing on both sides of the equation; -4 on both sides
SOLVE AN EQUATION OF THE FORM ax=b • Use the Multiplication Property of Equations to solve the unknown • Find the number that is on the same side of the equation as the unknown and multiply both sides by the reciprocal and the sign that is with the number. • Apply the Division Principle as a shortcut with integers and decimals when possible.
THE BASIC PERCENT EQUATION • Percent x Base = Amount • P x B =A • 20% of what number is 30? • Translate and solve the verbal expression. Change the percent to a decimal or fraction. • 0.20 x n = 30; Solve for n; 30 divided 0.20 = n; n = 150
MORE EXAMPLES OF THE BASIC PERCENT EQUATION • 70 is what percent of 80? • Translate and solve. Change the answer to a percent. • 70 = n x 80; 70 divided by 80 = 0.875 which is 87.5%; n = 87.5% • What is 40% of 80? • Translate and solve. Change the percent to a decimal. • 40% = 0.40;n = 0.40 x 80;n =32
3.2 GENERAL EQUATION PART 1 • Solve an equation of the form ax+b=c • The goal is to write the equation as variable=constant. • 5x+6=26 ; solve for x by applying the Addition Property of equations to +6 • 5x+6-6=26-6 • 5x=20; divide both sides by 5; x=4 • Check by replacing x with 4
3.3 GENERAL EQUATION PART 2 • To solve an equation of the form ax+b=cx+d • Apply the Addition Property of Equations twice and then the Multiplication Property of Equations to solve the unknown. • 7a-5=2a-20; subtract 2a from both sides ; 5a-5=-20; add 5 to both sides; 5a=-15; divide both sides by 5; a=-3 and check.
3.4 TRANSLATE AND SOLVE • Use the translation rules from chapter 2 and solve the equations of one variable. • Consecutive Integer Formulas • Consecutive Integers:n,n+1,n+2 • Consecutive Even Integers; n,n+2,n+4 • Consecutive Odd Integers; n,n+2,n+4
SUM OF TWO NUMBERS WORD PROBLEMS • Define the unknowns first. • Smaller number is x; Larger number is the sum minus x (the smaller number). • Translate and solve for the smaller number first and then the larger number. Each problem must have two answers and add to equal the original sum.
ADDITION AND SUBTRACTION OF POLYNOMIALS • Monomial- a polynomial of one term. • Binomial- a polynomial of two terms. • Trinomial- a polynomial of three terms. • Quadnomial- a polynomial of four terms • Descending Order- the exponents of the variable decrease from left to right in the answer.
4.1 ADDING AND SUBTRACTING POLYNOMIALS • Addition of polynomials • Combine the like terms inside both sets of parentheses(same sign-ADD; different signs-SUBTRACT). • Subtraction of polynomials • Multiply each term in the second polynomial by –1(there is a minus sign in front of the parenthese) then combine the like terms in both polynomials.
4.2 MULTIPLYING MONOMIALS • Multiply the coefficients and add the like variable exponents. • Simplifying powers of monomials • Distribute the outside exponent to each exponent in the monomial. Simplify the coefficient completely in the answer. This is the only time exponents are actually multiplied.
4.3 MULTIPLICATION OF POLYNOMIALS • Monomial times a polynomial. • Multiply the monomial by applying the distributive property to each term inside the parentheses( the polynomial) • Multiplying two polynomials. • Apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial and then combine the like terms. Place the answer un descending order.
TO MULTIPLY TWO BINOMIALS • Use the FOIL method to multiply two binomials. • This is the simple application of the distributive property in an ordered method. • F0IL METHOD;F- first terms are to be multiplied;O- outside terms are multiplied; I-inside terms are multiplied;L-last terms are multiplied. Combine the middle two terms if possible.
MULTIPLYING BINOMIALS WITH SPECIAL PRODUCTS • The Sum and Difference of two terms. • Do FOIL; the middle two terms will cancel; the answer will be a binomial with a minus sign between the terms. • The Square of a binomial. • Do FOIL; the middle two terms will be the same so add them; the answer will be a trinomial.
4.4 NEGATIVE EXPONENTS • Division of monomials. • To divide two monomials with the same base, subtract the smaller exponent from the larger exponent. • Zero as an exponent. • If zero is the dominant exponent the answer is always 1.
RULES FOR SIMPLIFYING NEGATIVE EXPONENTS • The negative exponent must be made positive by moving it to the opposite place in the fraction. This may be done first in the problem, but especially in the answer. • If there is a like base in the numerator and denominator and both exponents are negative they must be switched and made positive; then use division rules to simplify.
MIORE RULES FOR NEGATIVE EXPONENTS • If the bases are the same and one of the exponents is negative and one is positive, move the negative exponent to the positive exponent and ADD the exponents together. • When multiplying negative exponents, combine the like base`s exponents together using sign rules for addition and subtraction. Make neg. exponents positive.
SCIENTIFIC NOTATION • In scientific notation, a number is expressed as the product of two factors, the first number must be a number between one and ten(use of a decimal point may be needed), and the other number a power of ten. • To find the exponent in a number greater than one, count the place values after the first number.
MORE ON SCIENTIFIC NOTATION • To write a decimal in scientific notation. • Place a decimal point after the first number in the decimal. • To write the power of ten, count the place values from the decimal point to the first number in the decimal, this is the exponent.
4.5 DIVISION OF POLYNOMIALS • TO divide a polynomial by a monomial. • Divide each term of the polynomial(numerator) by the monomial.Simplify the expression.
TO DIVIDE POLYNOMIALS • The process for dividing polynomials is similar to the one for dividing whole numbers. The use of long division is the method. • Steps: Divide the like variable terms and place the answer in the quotient. Multiply the quotient by each term on the outside of the problem.
STEPS FOR DIVISION • Step 3 is to subtract the products( change the sign of the second term and combine the like terms). • The process starts over; divide, multiply, and subtract. • If there is a remainder, write it as a fraction.