Example: 15:21{\displaystyle 15:21} Note that neither number in this example is a prime number. Since that is the case, you’ll need to factor both numbers to determine whether or not the two terms have any identical factors that can cancel each other in the simplification process.
Note that neither number in this example is a prime number. Since that is the case, you’ll need to factor both numbers to determine whether or not the two terms have any identical factors that can cancel each other in the simplification process.
Example: The number 15 has four factors: 1,3,5,15{\displaystyle 1,3,5,15} 151=15{\displaystyle {\frac {15}{1}}=15} 153=5{\displaystyle {\frac {15}{3}}=5}
151=15{\displaystyle {\frac {15}{1}}=15} 153=5{\displaystyle {\frac {15}{3}}=5}
Example: The number 21 has four factors: 1, 3, 7, 21 211=21{\displaystyle {\frac {21}{1}}=21} 213=7{\displaystyle {\frac {21}{3}}=7}
211=21{\displaystyle {\frac {21}{1}}=21} 213=7{\displaystyle {\frac {21}{3}}=7}
Example: Both 15 and 21 share two common factors: 1 and 3 The GCF for the two terms of the original ratio is 3.
The GCF for the two terms of the original ratio is 3.
Example: Both 15 and 21 are divided by 3. 153=5{\displaystyle {\frac {15}{3}}=5} 213=7{\displaystyle {\frac {21}{3}}=7}
153=5{\displaystyle {\frac {15}{3}}=5} 213=7{\displaystyle {\frac {21}{3}}=7}
Example: 5:7{\displaystyle 5:7} The point of all this is that the simplified ratio 5:7 is easier to work with than the original ratio 15:21.
Example: 18x2:72x{\displaystyle 18x^{2}:72x}
Example: To solve this problem, you will need to find the factors of 18 and 72. The factors of 18 are: 1, 2, 3, 6, 9, 18 The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
The factors of 18 are: 1, 2, 3, 6, 9, 18 The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Example: Both 18 and 72 share several factors: 1, 2, 3, 6, 9, and 18. Of these factors, 18 is the greatest.
Example: Both 18 and 72 are now divided by the factor 18. 1818=1{\displaystyle {\frac {18}{18}}=1} 7218=4{\displaystyle {\frac {72}{18}}=4}
1818=1{\displaystyle {\frac {18}{18}}=1} 7218=4{\displaystyle {\frac {72}{18}}=4}
If there are exponents (powers) applied to the variable in both terms, deal with them now. If the exponents are the same in both terms, they cancel each other completely. If the exponents are not the same, subtract the smaller exponent from the larger. This completely cancels the variable with the smaller exponent and leaves the other variable with a diminished exponent. Understand that by subtracting one power from the other, you are essentially dividing the larger variable amount by the smaller one. Example: When examined separately, the ratio of variables was: x2:x{\displaystyle x^{2}:x} You can factor out an x{\displaystyle x} from both terms. The power of the first x{\displaystyle x} is 2, and the power of the second x{\displaystyle x} is 1. As such, one x{\displaystyle x} can be factored out from both terms. The first term will be left with one x{\displaystyle x}, and the second term will be left with no x{\displaystyle x}. x(x:1){\displaystyle x(x:1)} x:1{\displaystyle x:1}
You can factor out an x{\displaystyle x} from both terms. The power of the first x{\displaystyle x} is 2, and the power of the second x{\displaystyle x} is 1. As such, one x{\displaystyle x} can be factored out from both terms. The first term will be left with one x{\displaystyle x}, and the second term will be left with no x{\displaystyle x}. x(x:1){\displaystyle x(x:1)} x:1{\displaystyle x:1}
Example: The greatest common factor in this example is 18x{\displaystyle 18x}. 18x⋅(x:4){\displaystyle 18x\cdot (x:4)}
18x⋅(x:4){\displaystyle 18x\cdot (x:4)}
Example: x:4{\displaystyle x:4}
Example: (x2−8x+15):(x2−3x−10){\displaystyle (x^{2}-8x+15):(x^{2}-3x-10)}
Example: For this ratio you can use the decomposition method of factorization. x2−8x+15{\displaystyle x^{2}-8x+15} Multiply the a and c terms together: 1⋅15=15{\displaystyle 1\cdot 15=15} Find two numbers that equal this number when multiplied and add up to the value of the b term: −5,−3[−5⋅−3=15;−5+−3=−8]{\displaystyle -5,-3[-5\cdot -3=15;-5+-3=-8]} Substitute these two numbers into the original expression: x2−5x−3x+15{\displaystyle x^{2}-5x-3x+15} Factor by grouping: (x−3)⋅(x−5){\displaystyle (x-3)\cdot (x-5)}
x2−8x+15{\displaystyle x^{2}-8x+15} Multiply the a and c terms together: 1⋅15=15{\displaystyle 1\cdot 15=15} Find two numbers that equal this number when multiplied and add up to the value of the b term: −5,−3[−5⋅−3=15;−5+−3=−8]{\displaystyle -5,-3[-5\cdot -3=15;-5+-3=-8]} Substitute these two numbers into the original expression: x2−5x−3x+15{\displaystyle x^{2}-5x-3x+15} Factor by grouping: (x−3)⋅(x−5){\displaystyle (x-3)\cdot (x-5)}
Example: Use any method desired to break down the second expression into factors: x2−3x−10{\displaystyle x^{2}-3x-10} (x−5)⋅(x+2){\displaystyle (x-5)\cdot (x+2)}
Example: The factored form of the ratio is written as: [(x−3)(x−5)]:[(x−5)(x+2)]{\displaystyle [(x-3)(x-5)]:[(x-5)(x+2)]} The common factor in both terms is: (x−5){\displaystyle (x-5)} When the common factor is removed, the ratio can then be written as: [(x−3):(x+2)]{\displaystyle [(x-3):(x+2)]}
The common factor in both terms is: (x−5){\displaystyle (x-5)} When the common factor is removed, the ratio can then be written as: [(x−3):(x+2)]{\displaystyle [(x-3):(x+2)]}
Example: (x−3):(x+2){\displaystyle (x-3):(x+2)}
