Algebra 1: Key Concepts
This guide provides an overview of some key concepts related to the Algebra 1 course. It is not intended to cover every math concept addressed as part of the curriculum but does highlight major topics.
Number and Operations – perform operations with numbers and expressions to solve problems
Key Pearson Chapters: 1, 2, 3, 7, 8
Order of Operations: PEMDAS “Please Excuse My Dear Aunt Sally”:
1. Use Distributive Property to Eliminate Parentheses or Grouping Symbols
2. Evaluate all Exponents or Powers
3. Do all Multiplication and/or Division from left to right
4. Do all Addition and/or Subtraction from left to right
Square Roots and Cube Roots:
1. The area of a square is 25 m2. How long is each side? Since the area of a square is s2, s = √25 = 5 m.
** If you know the area of a square, take the square root to find the length of one side.
2. What is the cube root of 27 (3√27)? The cube root means that you can multiply a number by itself 3 times and get the number under the radical sign. So, what number when multiplied by itself 3 times gives you 27? Answer: 3 (because 3 x 3 x 3 = 27).
** If you know the volume of a cube, take the cube root to find the length of one side.
Fractional/Rational Exponents:
√x = x1/2
3√x = x1/3
4√x = x1/4
3√x2 = x(2)(1/3) = x2/3
n√xm = xm/n
Properties:
Additive Identity: a + 0 = 0 + a = a
Multiplicative Identity: a • 1 = 1 • a = a
Multiplicative Property of 0: a • 0 = 0 • a = 0
Multiplicative Inverse: (1/a)(a/1)= 1
Distributive Property: a(b + c) = ab + ac and a(b – c) = ab – ac and (b+ c)a = ba + ca and (b – c)a = ba - ca
Commutative Property: a + b = b + a and ab = ba
Associative Property: (a + b) + c = a + (b + c) and (ab)c = a(bc)
Like Terms: A term is a number, variable, or product/quotient of numbers and variables. Like terms are terms that contain the same variables with the same power. You must combine like terms and remove parentheses to have an expression in simplest form.
Examples:
3a2 + 5a2 = 8a2
4(3g + 2) = 12g + 8
6(5a + 3b – 2b) = 30a + 18b – 12b = 30a + 6b
Solving Linear Equations:
1. Use the Distributive Property to remove grouping symbols/parentheses
2. Simplify the expression on each side of the equals sign by combining any like terms
3. Use addition/subtraction to move all the variables on one side of the equals sign and all numbers without variables on the other side of the equals sign
4. Simplify the expressions on each side of the equals sign by combining any like terms
5. Use multiplication/division to get the variable by itself. Note: if the solution results in a false statement (3 ≠ 5), there is no solution; if the solution results in an identity (3 = 3), the solution is all numbers.
Example:
Solve 2(w – 3) + 5 = 3(w – 1)
Step 1: 2w – 6 + 5 = 3w – 3
Step 2: 2w – 1 = 3w – 3
Step 3: -w = -2 (subtracted 3w from both sides and added 1 to both sides to get variables on left and numbers on right)
Step 4: nothing to simplify
Step 5: w = 2 (divided both sides by -1 to get w by itself)
Inequalities: A statement using one of the 4 inequality symbols:
<: Less Than (Open circle on number line; Dashed boundary line)
≤: Less Than or Equal To (Closed circle on number line; Solid boundary line)
>: Greater Than (Open circle on number line; Dashed boundary line)
≥: Greater Than or Equal To (Closed circle on number line; Solid boundary line)
When you multiply or divide both sides of an inequality by a negative number, flip the inequality symbol.
Scientific Notation: Used to write very large or very small numbers in shortened form.
Step 1: Move the decimal point to the immediate RIGHT of the first NONZERO digit of the number.
Step 2: Multiply by a power of 10. If you moved the decimal point to the left, the exponent is positive. If you moved the decimal point to the right, the exponent is negative.
Examples: 345,000,000 = 3.45 x 108 and 0.0000000345 = 3.45 x 10-8
Absolute Value: the absolute value of a number is its distance from zero on the number line. The absolute value of a number is always positive. The absolute value is indicated by two vertical bars | |. Examples: |-9| = 9, |9| = 9, |5 – 8| = |-3| = 3
Set Notation (Section 3-5)
Roster Form: List the elements of a set within braces.
Example: Write “M is the set of integers greater than -3 and less than 4” in roster form. {-2, -1, 0, 1, 2, 3}
Set-Builder Notation: Describes the properties an element must have to be included in a set. {x|x > 2} is read “The set of all real numbers x, such that x is greater than 2”.
Compound Inequalities (Section 3-6)
Compound Inequality: 2 distinct inequalities joined by the words AND or OR.
And: A solution must be true for BOTH inequalities to be a solution to the compound inequality. The graph is only the OVERLAP/INTERSECTION of the 2 separate solution graphs.
4 < x < 12 à x > 4 AND x < 12 (exclusive: 4 and 12 are NOT included)
-3 ≤ x ≤ 2 à x ≥ -3 AND x ≤ 2 (inclusive: -3 and 2 ARE included)
OR: A solution only needs to be true for ONE of the inequalities to be a solution to the compound inequality. The graph is the COMBINATION/UNION of the 2 separate solution graphs.
Absolute Value Equations and Inequalities (Section 3-7)
1. Isolate the absolute value expression on one side of the equal sign.
2. Then, write and solve a pair of equations or inequalities.
3. Your solution will have 2 components … or no solution. Any equation that states that the absolute value of an expression is NEGATIVE has no solution.
Absolute Value Equations: To solve an equation in the form |A| = b, where A represents a variable expression and b>0, solve A = b and A = -b.
Example:
|r – 8| = 5
r – 8 = 5 r – 8 = -5
r = 13 r = 3
Solution: r = 13 OR r = 3
Absolute Value Inequalities:
|A| < b à -b < A < b à A > -b AND A < b
|A| ≤ b à -b ≤ A ≤ b à A ≥ -b AND A ≤ b
|A| > b à A < -b OR A > b
|A| ≥ b à A ≤ -b OR A ≥ b
|ax + c| < b / ax + c < b AND ax + c < -b|ax + c| ≤ b / ax + c ≤ b AND ax + c ≥ -b
|ax + c| > b / ax + c > b OR ax + c < -b
|ax + c| ≥ b / ax + c ≥ b OR ax + c ≤ -b
Examples:
Solve |2c – 5| < 9
2c – 5 < 9 AND 2c – 5 < -9
2c < 14 2c < -4
c < 7 c < -2
Solution: c < 7 AND c < - 2 à -2 < c < 7
Solve |y + 8| > 3
y + 8 > 3 OR y + 8 < -3
y > -5 y < - 11
Solution: y < -11 OR y > -5
Fractions, Decimals, and Percents:
To convert a fraction to a decimal, divide the numerator by the denominator.
To convert a decimal to a percent, multiply by 100 and add the % symbol.
To convert a percent to a decimal, divide by 100 and remove the % symbol.
Percent of Change:
To find a percent of change: (new – original) (100)
original
Polynomials (Chapters 7 and 8):
Monomial:
· A number, a variable, or a product of a number and one or more variables.
· An expression involving the division of variables is NOT a monomial. Expressions with variables raised to negative or fractional exponents are NOT monomials
· Monomials that are real numbers are called constants.
Product of Powers:
Product of Powers = Multiplying Powers:
Words: To multiply 2 powers that have the same base, add the exponents
Symbols: For any number a and all integers m and n: am · an = a m+n
Example: a4 · a12 = a 4+12 = a 16
Power of a Power:
Words: To find the power of a power, multiply the exponents.
Symbols: For any number a and all integers m and n: (am)n = am·n
Example: (k5)9 = k5·9= k45
Power of a Product:
Words: To find the power of a product, find the power of each factor and multiply.
Symbols: For all numbers a and b and any integer m: (ab)m = ambm
Example: (-2xy)3 = (-2)3x3y3 = -8x3y3
Quotient of Powers:
Words: To divide two powers that have the same base, subtract the exponents.
Symbols: For all integers m and n and any nonzero number a, am/an = am-n
Example: b15/b7 = b15-7 = b8
Power of a Quotient:
Words: To find the power of a quotient, find the power of the numerator and the power of the denominator.
Symbols: For any integer m and any real numbers a and b (b cannot equal zero): (a/b)m = am/bm
Example: (c/d)5 = c5/d5
Zero Exponent:
Words: Any nonzero number raised to the zero power is 1.
Symbols: For any nonzero number a, a0 =1
Example: (-0.25)0 = 1
Negative Exponent:
Words: For any nonzero number a and any integer n, a-n is the reciprocal of an. In addition, the reciprocal of a-n is an .
Symbols: For any nonzero number a and any integer n, a-n = 1/an and 1/a-n = an
Example: 5-2 = 1/52 or 1/25
Name of Rule / Symbols / ExampleProduct Rule / am · an = am+n / x5· x2 = x7
Quotient Rule / am/an = am-n / x5/ x2 = x3
Power of a Product Rule / (ab)m = ambm / (3xy)2 =32x2y2 =9x2y2
Power of a Quotient Rule / (a/b)m = am/bm / (3x/4y)2 = 9x2/16y2
Power of a Power Rule / (am)n = amn / (x5)2 = x10
Zero Power Rule / a0 = 1 / (x3y5z12)0 = 1
Negative Exponents Rule 1 / a-m = 1/am / x-5 = 1/x5
Negative Exponents Rule 2 / 1/a-m = am / 1/x-2 = x2
Negative Exponents Rule 3 / (a/b)-m = (b/a)m / (2x/y)-2 = y2/4x2
Rational Powers / n√xm = xm/n / 4√x3 = x3/4
Polynomial:
· A polynomial is a monomial or sum of monomials.
· Binomial: sum of 2 monomials
· Trinomial: sum of 3 monomials
Degree of a monomial: The degree of a monomial is the sum of the exponents of all of its variables.
Degree of a polynomial: The degree of a polynomial is the greatest degree of any term in the polynomial. To find the degree of a polynomial, find the degree of each monomial and then choose the greatest degree.
Dividing Monomials:
To divide 2 powers that have the same base, subtract the exponents
Example: (3a2bc)/(12ab2) = ac/4b
Adding and Subtracting Polynomials:
To add polynomials: group then combine like terms
Example: (3 + a2 + 2a) + (a2 – 8a + 5) = (a2 + a2) + (2a – 8a) + (3 + 5) = 2a2 – 6a + 8
To subtract polynomials: Use the additive inverse (flip the signs) of the polynomial being subtracted
Example: (11 + 4d2) – (3 – 6d2) = 11 + 4d2 – 3 + 6d2 (flip the signs of the polynomial that followed the subtraction sign) = (4d2 + 6d2) + (11 – 3) = 10d2 + 8
Multiplying Binomials: Use FOIL (First, Outer, Inner, Last) then combine like terms
Example: (2y + 3)(6y - 7) = 12y2 – 14y + 18y – 21 = 12y2 + 4y – 21
Multiplying Polynomials: Use the Distributive Property then combine like terms
Example: (p + 4)(p2 + 2p – 7) = p3 + 2p2 – 7p + 4p2 + 8p – 28 = p3 + 6p2 + p – 28
Special Products:
Square of a Sum: (a + b)2 = a2 + 2ab + b2
Square of a Difference: (a - b)2 = a2 - 2ab + b2
Product of a Sum and Difference: (a + b)(a – b) = a2 – b2
Geometry and Measurement – Describe geometric figures in the coordinate plane algebraically.
Simplifying Radicals (Section 10-2):
To simplify a radical, break the expression under the radical sign into prime factors and then “pull out” any pairs (perfect squares).
Example: √12 = √2*2*3 = 2√3
A simplified radical cannot have a radical in the denominator. We solve this by “rationalizing the denominator”.
Example:
√10 = √10 * √12 = √2*5*2*2*3 = 2√30 = √30
√12 √12 √12 12 12 6
Operations with Radical Expressions (Section 10-3):
Radicals can be added or subtracted like monomials – think of each “radical” as a different variable.
Examples: √18 + √8 = √3*3*2 + √2*2*2 = 3√2 + 2√2 = 5√2
2√3 * 5√27 = 10√81 = 10√3*3*3*3 = 10*3*3 = 90
Midpoint Formula (Page 605):
The midpoint between 2 points (x1, y1) and (x2, y2) is a new ordered pair : ((x1+ x2)/2, (y1 + y2)/2)
New x-coordinate: sum of the 2 x-values divided by 2
New y-coordinate: sum of the 2 y-values divided by 2.
Example: Find the midpoint between (4, 2) and (8, 4). Using the above formula, the midpoint would be (6, 3).
Distance Formula (Page 605): The distance between 2 points (x1, y1) and (x2, y2) is given by the formula:
Pythagorean Theorem (Section 10-1): Can be used to find the lengths of right triangles.
a2 + b2 = c2 where a and b are the legs of the right triangle and h is the hypotenuse of the right triangle.
If a2 + b2 ≠ c2, then the triangle is NOT a right triangle.
Scale Factors and Dilations:
A dilation is when a figure is made bigger or smaller using a scale factor. If the scale factor is between 0 and 1, the new figure will be smaller than the original. If the scale factor is greater than 1, the new figure will be larger than the original.
When a scale factor is provided, you multiply the x and y coordinates by that number to get the new coordinates.
Example: A triangle has the following vertices (-1, 1), (6, -2), (3, 5). If the triangle undergoes a dilation with a scale factor of 3, what will be the vertices of the image? To solve, multiply each coordinate by the scale factor of 3: