LCM The least common multiple is the smallest number that is a multiple of both numbers. (not zero!) Prime factorization can be used to determine the least common multiple. Prime factorize all numbers Take all factors and multiply them together to get LCM (If you have a duplicates of factors you just take the one with the bigger power Ex. you have one 2^2 and one 2^4, take 2^4) GCF The least common multiple is the smallest number that is a multiple of both numbers. (not zero!) Prime factorization can be used to determine the least common multiple. Prime factorize all numbers Take all factors and multiply them together to get LCM (If you have a duplicates of factors you just take the one with the bigger power Ex. you have one 2^2 and one 2^4, take 2^4) Prime factorization Prime factorization is the process of finding which prime numbers multiply to the original number. Square & cube roots How to check if something is a perfect root of x: 1. Prime factorize the radicand and write it out 2. If all the exponents match the index of the radical then it's a perfect root for that index,x 3. Cancel out the powers and remove the radical then multiply what you have left to find the answer Ex. Check if ∛5832 1. Prime factorize: 5832 = 2^3 x 3^6 2. Check if exponents of factors in prime factorization are multiplies of the index: yes 3. That means this 5832 is a perfect cube, now to find the answer just simplify: ∛2^3x3^6 = 2 x 3^2 = 18 Prime numbers - A number that has only two factors, 1, and itself Note* - 1 is not a prime number Composite numbers - A number greater than 1 that’s NOT prime. All composite numbers can be written as a product of primes. Divisibility rules 0 0 is divisible by everything, nothing is divisible by 0. 1 Everything is divisible by one 2 All even numbers are divisible by 2 3 If the sum of all digits is divisible by 3 4 If the last two digits as a number divisible by 4 5 Last digit is 0 or 5 6 If the number is divisible by 2 and 3 7 If the difference of the last digit doubled minus the rest of the digits as a number is divisible by 7 8 If the last three digits as a number is divisible by 8 If the number is a three or less digit number then if it’s divisible by 4 it’s also divisible by 8 (except 4) 9 If the sum of the digits is divisible by 9 10 If the number ends in 0 11 If the alternating difference of the digits is divisible by 11 12 If the number is also divisible by 3 & 4 Exponent laws http://people.sunyulster.edu/nicholsm/webct/WebCT2/laws_of_exponents.htm Absolute Value The distance of a number from zero Lesson 4 - Intro to factoring ax^2+bx+c Lesson 5 -Factoring quadratics & factoring by grouping Lesson 6 - Factoring through decomposition Decomposition - a way to factor, when used the answer will always be true I use it when ‘a’ and ‘c’ are both greater than 1 Steps when factoring through decomposition: 1. Take out GCF (if possible) 2. Multiply ‘a’ & ‘c 3. Find two numbers that will multiply into ac, and have a sum of b 4. Rewrite ‘bx’ as a sum 5. Factor through grouping 6. Double check to see if anything is still factorable Ex. Factor 6x^2+3x-4 6x-4=-24, 3 & -8 multiply to -24, and have a sum of 3, rewrite: 6x^2+3x-8x-4, factor through grouping: 3x(2x+1)-4(3x+1) = (3x-4)(2x+1) Lesson 7 - factoring difference of squares Factoring difference of squares using conjugates When you encounter something in the form of ax-by & all of them are perfect squares, then you can factor it by multiplying it with its conjugate Ex. x^2-4 = (x-2)(x+2), x^2y^2=(x-y)(x+y) You cannot factor a sum of squares Factoring with fractions Factor 4a^2+2a+⅖ 1. Undistribute ⅖ , to do so divide all terms by numerator, and divide by the denominator: ⅖(10a^2+5a+1) 2. If possible factor more Factor x^3+⅕x^2-⅓x 1. Undistribute the LCM of the fractions, in this case the LCM of ⅕ & ⅓ is 1/15 2. Divide by numerator and multiply by denominator: 1/15x(15x^2+3x-5) 3. Factor more in possible Lesson 8 - Factoring special cases Perfect square trinomials (a+b)^2 = a^2+2ab+b^2 (a-b)^2= a^2-ab+b^2 Factoring a[f(x)]^2=bf(x)+c Can only use if the exponents are going down in a geometric sequence between each monomial Factor x^4-5x^2-14 Check if the powers are greater than 2, then assign a variable to the high power Let A=x^2 Replace & factor: A^2-5A-14 = (A-7)(A+2) Substitute values: (x^2-7)(x^2+2) Lesson 9 - Factoring a sum & difference of cubes Sum of cubes a^3+b^3 = (a+b)(a^2-ab+b^2) Difference of cubes a^3-b^3= (a-b)(a^2+ab+b^2) How to determine what method to use when factoring? 1. Take out a GCF (If possible) 2. Two terms? perfect square trinomials (a^2-b^2), or sum/difference of cubes (a^3±b^3) Three terms? Inspection if a is 1, or Decomposition if a & c are > than 1, or Perfect square trinomial (a+b)^2 = a^2+2ab+b^2 (a-b)^2= a^2-ab+b^2, or a[f(x)]^2=bf(x)+c if the exponents are big & are a geometric sequence Four or more terms? Grouping Check if there’s a perfect square trinomial w/3 of the terms? 3. Double check, check for more GCF’s or if you can further factor Polynomials Degree of polynomials - the monomial which has the biggest degree of the polynomial is the polynomials degree Q. Determine the degree: 7x^2+5x-6: degree 2 polynomial 3xy^2+16x^3y^3+17x^3y^4: degree 4 polynomial (quadratic) When dividing polynomials if you aren’t dividing by a monomial you can use long division Long division https://www.youtube.com/watch?v=_FSXJmESFmQ Synthetic division https://www.youtube.com/watch?v=FxHWoUOq2iQ ● Word Problems (LCM/GCF)
