ALGEBRA
What is an equation
A mathematical sentence that states on quantity is equal to another.
Multiplicative inverse Rule
A multiplicative inverse is a reciprocal. What is a reciprocal? A reciprocal is one of a pair of numbers that when multiplied with another number equals the number 1. For example, if we have the number 7, the multiplicative inverse, or reciprocal, would be 1/7 because when you multiply 7 and 1/7 together, you get 1!
Negative exponent
A negative exponent means how many times to divide by the number. Example: 8-1 = 1 ÷ 8 = 1/8 = 0.125. Or many divides: Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008. Flip the base upside down and make the power positive. In a the fraction 3/8 negative 3 power, flip it to 8/3 positive 3 power or cubed. If whole number, add a one on top and make the power positive.
Composite Number
A number grater than one which is not prime. The first 10 composite numbers are 2, 6, 8, 9, 10, 12, 14, 15, 16, 18, and 20.
Factor
A number that divides equally into another number . Example 1, 2, 3, 4, 5 and 12 are all factors of 12. 12 has 6 factors.
Power of a Power Property
A power raised to another power is the based raided to the product of the powers. Multiply the inside power of the parathenthesis by the outside power.
What is proportion?
A proportion is an equation stating that two ratios are equal.. ex: 3/12 = 9/36 is a proportion
Power of a quotient property
A quotient raised to the power is the quotient of the powers.
What is a ratio?
A ratio compares two quantities by division. A ratio is a fraction.
Perimeter of a Triangle
Add together all three sides of Triangle or a+b+c=perimeter.
What is the addition property of equality?
Additive Property of Equality. The formal name for the property of equality that allows one to add the same quantity to both sides of an equation. Adding or subtracting the same property to both sides of the equation will not change the solution.
Polynomial
An expression that is the sum or difference of two or more terms
Irrational number
An irrational number has endless non-repeating digits to the right of the decimal point. Square roots of non-perfect squares.
Zero Exponents
Anything raised to the zero power is 1.
What is Multiplication and division of property of equality?
Lets you multiply or divide both sides of an equation by the same number.
Area of a square
Measured in Units or U squared. For find the area of a square, multiply the length by the width.
Multiplying Monomials
Monomials can be multiplied or divided regardless of whether they are similare or not.
Multiply Fractions
Multiply each numerator or top number and each denominator or bottom number together, then simplify if possible.
Multiplicative Identity (Multiplying a number times one)
Multiplying any number by one does not change it's value.
Integers
Natural numbers and their opposites. Subset of Rational numbers
Adding & Subtracting Monomials
Need to have the same variable and same power
Can you change the order of number in subtraction with changing the answer?
No order matters!
Opposite numbers
Numbers that are the same distance from zero on a number line, but in opposite directions from zero. Opposites have the same value, when added together, the sum is always zero
Rules for multiplying integers or Natural numbers and their opposites.
Numbers with the same sign, multiply their absolute values, if the same is the same whether it is negative or positive, the answer or product will be positive. Two negative signs will equal a positive product or answer.
Monomial
One term
Order of Operations
PEMDAS -P= Parenthesis/Grouping Symbols, E=Expomemts, M=Multiplication, D=Division, A=Addition, S=Subtraction.
Percent
Per 100, represents a ratio in terms of 100.
Rectangle perimeter
Perimeter equals Length plus length plus width plus width. Also 2L Plus 2w
Difference of squared
Product of two binomials whose terms are the same by have different signs.
Division
Quotient, per, ratio, shared equally, average, ratio, Examples, How much for each? How many in each group? How much per year or person? The ratio of 4 to 2.
Real Numbers
Real numbers can be graphed on a number line Positive or negative, large or small, whole numbers or decimal numbers and fractions are all Real Numbers.
Rules for dividing integers
Same for both division and multiplication. Same sign, quotient or answer is positive. Different sign the quotient or answer is negative.
Rules for adding integers or natural numbers and their opposites.
Same sign rule - just add the number and keep the sign. Mixed sign rule subtract and keep the sign of the larger number.
Addition
Sum, plus, and, also, greater, larger, in excess of
DIVIDING FRACTIONS
TO Divide fractions, multiply by the reciprocal and divide out common factors if possible. It is only completely simplified when all common factors have be divided by itself. Flip the 2nd Fraction and multiply.
Absolute Value
The absolute of a real number also known as Natural numbers and their opposites is its distance from zero on the number line.
Rectangle area
The area of a equals multiply the length by the width or L(w).
Natural Whole numbers
The counting numbers and zero. Subset of intergers. Does not include zero. Does not include negative numbers. Does not include fractions (such as 1/2 or 3/7) Does not include decimals
Perimeter
The distance around a figure or the sum -total added together - of all sides.
Distributive Property
The distributive property lets you multiply a sum by multiplying each addend separately and then adding the products. When you distribute you are expanding and breaking into parts.
Distributing a Negatives in algebra
The negative sign applied to a group something in parantheses changes the sign of each term in the group.
Commutative Property of Addition
The order in which the numbers are added does not change the sum.
Prime Factorization
The process of breaking a composite number into the product of prime factors.
When multiplying integers or Natural numbers and their opposites that have DIFFERENT signs
The product or answer will always be NEGATIVE
Additive Inverse Property - adding any positive number to its identical negative number.
The sum of any number and it's opposite is zero. For instance, 7 plus negative 7 equals zero or 7 +(-7) = 0
Multiplying negative numbers with exponents
The use of parentheses when multiplying exponents can change the sign of your product.
Associative Property of Addition
The way the numbers are grouped does not change the sum.
Divide Monomials
To divide a monomial by a monomial, divide the coefficients (or simplify them as you would a fraction) and divide the variables with like bases by subtracting their exponents.
Product of a Power Property
To multiply two expressions with the same base, add the exponents and keep the base.
Multiplying Fractions
Two methods 1. Multiply straight across then cancel out/divide common factors. 2. Divide out common factors first diagonally, then multiply straight across and simplify.
Like Terms
Two or more terms wit the same Variable part. Like terms can be added together. Add the numbers and carry the variable.
Multiplying binomial by polynomial
Use Distributive property
Whole Numbers
Whole numbers are positive numbers, including zero, without any decimal or fractional parts. They are numbers that represent whole things without pieces.
Properties of division and zero
Zero divided by any non-zero number is zero. Division by zero is undefined. If zero is under a number, the answer is UNDEFINED.
Additive Identity Property - zero rule. Nothing gained or lost by adding zero.
Zero is unique, if you add zero to a number it give you the identical number with nothing gained or lost.
3rd degree cubic
cubic: a third-degree polynomial, such as -6x3 or x3 - 27 (because the variable in the leading term is cubed
Multiplication
of, product, times, factors, squared, cubed, Examples One-half of twenty, ten percent of the total, double the order, product of 5 and 5.
2nd Degree quadratic
quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, second degree polynomial equation since the greatest power is two.
Solving an equation with a variable on both sides
1. Combine like terms and clear fractions by multiply both sides by the LCD. 2. Get all your variable terms to one side of the equation by adding or subtracting the smaller variable term so you son have to deal with negative coefficients. 3. Add or subtract the constant term from the other side. 4. Divide to find the value of your variable.
Solving equations when there is a variable on both sides.
1. Simplify by distributing or combining like terms. Clean fractions by multiplying both sides by LCD. 2. Get all variable terms to one side of the equation. Add or subtract the smaller variable term so you don't have to deal with negative coefficients. 3. Add or subtract the constant term from the other side. 4. Divide to find the value of the variable.
Perfect square trinomial
A Squared + 2ab + b squared Since all signs are positive, the pattern is (a + b)squared= a squared + 2ab + b squared.
Dividing fractions
the same as multiplication by its reciprocal.
Any Negative number divided by itself is always 1
-9 divided by -9 equals 1
Subtracting Real numbers
Can be thought of as adding the opposite.
Associative property of multiplication
Changing the grouping of the numbers in a product will not change the result.
Commutative Property of Multiplication
Changing the order of the numbers will not change the result.
Formula
Contains one or more variables
1st degree linear polynomial
Degree 1 polynomials are called linear polynomials because their graphs are straight lines.
Subtraction
Difference, less, less than, decrease, diminished
Vocabulary for Division
Dividend - Dividing into, Divisor - divided by, quotient is the answer. Also Divisor times quotient equals the dividend.
Area of a Triangle
Equals half of the base times height.
Order of Operations containing both multiplication and division or addition and subtraction
Evaluate from left to right
Two terms with different addition or subtraction signs.
Even if the problem has a plus sign when you have different signs you must subtract and keep the sign of the larger number. For instance negative 25 plus ten. You drop the all the sign and just subtract 10 from 25. Then you keep the negative sign that was on the 25 because 25 was the larger number.
When there is a minus (-) sign in front of parenthesis change the sign of each tern in the parenthesis. 5 minus paraenthesis 7x plus 4 becomes 7x minus 4, then subtract the 4 from the 5 because they are like terms. Final answer 1 minus 7X
Final answer 1 minus 7X
Foil Method
First, Outer, Inner, Last
Rational number
Fractions or decimals that do terminate or repeat.
Equivalent Fractions
Fractions that have the same value such as 1/2 and 2/4 or 1/3 and 3/9.
Prime Number
Has only 2 factors, itself and one. The 1st 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41.
Variable
In Math, a variable is an unknown quantity, a letter represents the variable or unknown quantity.
