Mathematics college algebra formula sheet, Cheat Sheet of Algebra

Download Mathematics college algebra formula sheet and more Cheat Sheet Algebra in PDF only on Docsity! MATH 133 College Algebra Polynomials: 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ 𝑎2𝑥2 + 𝑎1𝑥 + 𝑎0 Standard form: a polynomial written with descending powers of the variable.  Monomial: polynomial with one term 5𝑥  Binomial: polynomial with two unlike terms 10𝑥2 + 5𝑥  Trinomial: polynomial with three unlike terms 10𝑥2 + 5𝑥 + 4 Degree of the polynomial is the highest powered variable. Ex. – 𝑥7 + 16𝑥5 + 5𝑥2 − 4𝑥 + 12 Degree = 7. Polynomial Long Division: 3𝑥3 + 4𝑥2 + 𝑥 + 7 ÷ 𝑥2 + 1 3𝑥 + 4 7431 232  xxxx −(3𝑥3 + 3𝑥) 4𝑥2 − 2𝑥 + 7 −(4𝑥2 + 4) −2𝑥 − 3 1. Divide the first term in the divisor by the first term in the dividend. Then multiply the result with the second term in the divisor. 2. Subtract and bring down the next term. 3. Repeat until you can’t divide any more. Synthetic Division: Can be used as a shortcut when a polynomial is divided by 𝑥 − 𝑐 2𝑥3 − 𝑥2 + 3 ÷ 𝑥 − 3 3⌋ 2 − 1 + 0 + 3 ↓ +6 + 15 + 45 2 + 5 + 15 + ⌊48 2𝑥2 + 5𝑥 + 15 + 48 𝑥−3 1. Use the inverse of “c” as the divisor, and write only the coefficients of the polynomial. Don’t forget placeholders for missing variable. 2. First number comes straight down. 3. Multiply that number and the divisor together, and place under the next number. 4. Add the two numbers together. Repeat until there are no numbers left. 5. If you end up with a number other than zero, this will be your remainder. 6. When rewriting, begin with 1 degree less than when you started. Rational Exponents: 𝑎 1 𝑛 = √𝑎 𝑛 𝑎 𝑚 𝑛 = ( √𝑎 𝑛 ) 𝑚 Product and Quotient Rules for Radicals: √𝑢𝑣 𝑛 = √𝑢 𝑛 √𝑣 𝑛 √ 𝑢 𝑣 𝑛 = √𝑢 𝑛 √𝑣 𝑛 Radicals and Roots: √𝑎 𝑛  If “a” is a positive real number and “n” is even, then “a” has exactly two real nth roots Ex. √81 = ±9  If “a” is any real number and “n” is odd, then “a” has only one real root Ex. √27 3 =3  If “a” is a negative real number and “n” is even, then “a” has no real root. Ex. √−64 Rationalizing the Denominator: √ 3 5 = √3 √5 × √5 √5 = √15 5 4 √9 3 = 4 √9 3 × √3 3 √3 3 = 4√3 3 √27 3 = 4√3 3 3 Adding and Subtracting Radicals: When adding or subtracting radicals, the number under the radical and root must be the same, just like when combining variables. Ex. √3 + √3 = 2√3 (Remainder is written over the divisor.) Multiplying and Dividing Radicals: When multiplying and dividing radicals, do so just like you would variables. Ex. 3√2 × 5√6 = 15√12 Complex Numbers: In order to take square roots of negative number, the number 𝑖 is used. 𝑖 = √−1 Standard form: 𝑖2 = −1 𝑎 + 𝑏𝑖 𝑖3 = −𝑖 The conjugate of 𝑎 + 𝑏𝑖 is 𝑎 − 𝑏𝑖. 𝑖4 = 1 This is important for solving equations with complex numbers. Distance Formula: Finding the distance between two points (𝑥1, 𝑦1) and (𝑥2, 𝑦2). 𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Ex. (1,2) (5,9) 𝑑 = √(5 − 1)2 + (9 − 2)2 𝑑 = √42 + 72 𝑑 = √16 + 49 𝑑 = √65 ≈ 8.06 Midpoint Formula: Finding the midpoint of (𝑥1, 𝑦1) and (𝑥2, 𝑦2). 𝑀 = ( 𝑥1 + 𝑥2 2 , 𝑦1 + 𝑦2 2 ) Ex. (1,5) (4,7) 𝑀 = ( 1 + 4 2 , 5 + 7 2 ) 𝑀 = ( 5 2 , 12 2 ) ≈ ( 5 2 , 6) Sketching a Graph: 1. Build an xy table. 2. Plug in at least 3 values for x (You might need more than 3 values) 3. Find the y-values. 4. Plot the points Draw the line through those points. Discriminant of a Quadratic Equation:  If 𝑏2 − 4𝑎𝑐 > 0, two unequal real solutions.  If 𝑏2 − 4𝑎𝑐 = 0, a repeated real solution with a root of multiplicity 2.  If 𝑏2 − 4𝑎𝑐 < 0, no rea solution. Useful Equations:  Interest 𝐼 = 𝑝𝑟𝑡 𝐼 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙(𝑟𝑎𝑡𝑒)(𝑡𝑖𝑚𝑒)  Distance 𝑑 = 𝑟𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑟𝑎𝑡𝑒(𝑡𝑖𝑚𝑒)  Compound Interest 𝐴 = 𝑃 (1 + 𝑟 𝑛 ) 𝑛𝑡  Continuous Compounding Interest 𝐴 = 𝑃𝑒𝑟𝑡 Inequality Intervals: [𝑎, 𝑏] 𝑎 ≤ 𝑥 ≤ 𝑏 a[ ]b (𝑎, 𝑏] 𝑎 < 𝑥 ≤ 𝑏 a( ]b [𝑎, 𝑏) 𝑎 ≤ 𝑥 < 𝑏 a[ )b (𝑎, 𝑏) 𝑎 < 𝑥 < 𝑏 a( )b [𝑎, ∞) 𝑥 ≥ 𝑎 a[ (𝑎, ∞) 𝑥 > 𝑎 a( (−∞, 𝑏] 𝑥 ≤ 𝑏 ]b (−∞, 𝑏) 𝑥 < 𝑏 )b (−∞, ∞) -∞ ∞ Solving Absolute Value Equations: Absolute value equations can be split into two equations. Ex. |4𝑥 + 5| = 17 4𝑥 + 5 = 17 4𝑥 + 5 = −17 |𝑥 + 1| > 5 𝑥 + 1 > 5 𝑥 + 1 < −5 Absolute Value: This |𝑎| denotes absolute value, which is the distance a number is from the origin, 0, on the number line. Ex. |1| = 1 |−1| = 1 Solving Inequalities: Treat {<, >, ≤, ≥} like {=}. Except flip the sign when × 𝑜𝑟 ÷ by a negative number. Ex. 4 − 𝑥 ≥ 5 → −𝑥 ≥ 1 → 𝑥 ≤ −1 Symmetry:  With respect to x-axis (𝑥, 𝑦) 𝑎𝑛𝑑 (𝑥, −𝑦) 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ  With respect to y-axis (𝑥, 𝑦) 𝑎𝑛𝑑 (−𝑥, 𝑦) 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ  With respect to the origin (𝑥, 𝑦) 𝑎𝑛𝑑 (−𝑥, −𝑦) 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ Horizontal line test: Determines a one-to-one function like the vertical line test. Inverse Function: First must be one-to-one. Definition: 𝑓(𝑓−1(𝑥)) = 𝑥 𝑓−1(𝑓(𝑥)) = 𝑥 Given 𝑓(𝑥) = 𝑎𝑥 + 𝑏, replace 𝑓(𝑥) with 𝑦 and interchange 𝑥 and 𝑦. Solve for 𝑦. Ex. 𝑓(𝑥) = 3𝑥 + 5 𝑦 = 3𝑥 + 5 𝑥 = 3𝑦 + 5 𝑥 − 5 = 3𝑦 𝑥 − 5 3 = 𝑦 Exponential Function: 𝑓(𝑥) = 𝐶𝑎𝑥 𝑒 is irrational and often the base of the exponential function, but not always. If 𝑎𝑣 = 𝑎𝑢, then 𝑣 = 𝑢 Uninhibited growth/decay: 𝐴(𝑡) = 𝐴0𝑒𝑘𝑡 𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡 = 𝑡𝑖𝑚𝑒 𝐴0 = 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝑡 = 0 Logarithmic functions: 𝑦 = log𝑎 𝑥 if and only if 𝑥 = 𝑎𝑦 𝑦 = 𝑙𝑛𝑥 if and only if 𝑥 = 𝑒𝑦 𝑦 = log 𝑥 if and only if 𝑥 = 10^𝑦 Systems of Equations: Two equations with two unknowns. Can be solved two ways 1. Elmination { 4𝑥 − 5𝑦 = 13 3𝑥 − 𝑦 = 7 - { 4𝑥 − 5𝑦 = 13 −15𝑥 + 5𝑦 = −35 −11𝑥 = −22 𝑥 = 2 2. Substitution { 𝑥 + 𝑦 = 3 2𝑥 − 𝑦 = 0 { 𝑥 = 3 − 𝑦 2𝑥 − 𝑦 = 0 2(3 − 𝑦) − 𝑦 = 0 Properties of logarithms: log𝑎(𝑀𝑁) = log𝑎 𝑀 + log𝑎 𝑁 log𝑎 𝑀 𝑁 = log𝑎 𝑀 − log𝑎 𝑁 log𝑎 𝑀𝑟 = 𝑟 log𝑎 𝑀 𝑎𝑥 = 𝑒𝑥 ln 𝑒 ln 𝑒 = 1 𝑒ln = 1 log𝑎 𝑎 = 1 log𝑎 1 = 0 4(2) − 5𝑦 = 13 8 − 5𝑦 = 13 −5𝑦 = 5 𝑦 = −1 6 − 2𝑦 − 𝑦 = 0 −3𝑦 = −6 𝑦 = 2 𝑥 = 3 − 2 𝑥 = 1 Quadratics and Factoring Standard Form: 𝐴𝑥2 + 𝐵𝑥 + 𝐶 = 0 X-Factoring Method 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑎𝑥2 + 𝑚𝑥 + 𝑛𝑥 + 𝑐 = 0 (𝑎𝑥2 + 𝑚𝑥)(+𝑛𝑥 + 𝑐) = 0 Factor out common factor and solve for ‘x’. Perfect Square Trinomial 𝑎2 + 2𝑎𝑏 + 𝑏2 = 0 (𝑎 − 𝑏)2 = 0 𝑎2 − 2𝑎𝑏 + 𝑏2 = 0 (𝑎 − 𝑏)2 = 0 Quadratic Formula 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑥 = −𝑏 ± √𝑏2 − 4𝑎𝑐 2𝑎 Difference of two squares (𝑎2 − 𝑏2) = 0 (𝑎 + 𝑏)(𝑎 − 𝑐) = 0 Completing the Square 𝑎𝑥2 + 𝑏𝑥 = 𝑑 𝑎𝑥2 + 𝑏𝑥 + ( 𝑏 2 ) 2 = 𝑑 + ( 𝑏 2 ) 2 (√𝑎 𝑥 + 𝑏 2 ) 2 = 𝑑 + ( 𝑏 2 ) 2 √(√𝑎𝑥 + 𝑏 2 ) 2 = √𝑑 + ( 𝑏 2 ) 2 √𝑎𝑥 + 𝑏 2 = ±√𝑑 + 𝑏 2 Sum or Difference of Two Cubes (𝑎3 − 𝑏3) (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2) (𝑎3 + 𝑏3) (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2) Square Root Method (𝑎 + 𝑏)2 = 𝑐2 √(𝑎 + 𝑏)2 = √𝑐2 (𝑎 + 𝑏) = ±𝑐 𝑎 + 𝑏 = 𝑐 𝑎 + 𝑏 = −𝑐 𝑎 ∗ 𝑐 𝑏 𝑚 𝑛 (𝑚 + 𝑛) = 𝑏 (𝑚 ∗ 𝑛) = (𝑎 ∗ 𝑐)