Expansion - done Flashcards
What is the pattern with binomial expansion
The final result contains four terms:
(a + b) (c + d) = ac + ad + bc + bd
1. is the product to the First terms of each bracket
2. is the product to the Outer terms of each bracket
3. is the product of the Inner terms of each bracket
4. is the product of the Last terms of each bracket
What is the pattern with difference of squares expansion
(a+b) (a-b) = a^2 - b^2
The middle terms end up cancelling each other out
When do we use binomial expansion
When there is a term in front of a bracket, we treat it as a coefficient and can expand the brackets by using distributive law. When the coefficient is another bracket, we use FOIL/Feeding the chickens, or binomial expansion.
What is the pattern with perfect square expansion
(a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2
(a-b)^2 = (a-b)(a-b) = a^2 - 2ab +b^2
The middle term is twice the product (the number you get when you multiply) of the terms.
We FOIL/Feed the Chickens just as we do with regular expansion.
How does further expansion work (more than two numbers in a bracket)
(a+b)(c+d+e) = (ac)+(ad)+(ae)+(bc)+(bd)+(be)
We can still FOIL/Feed the chickens, just with more terms
What is the difference between a radical and a surd?
Radical: a number that is expressed using a √ , which is rational
Surd: a number that is expressed using a √ sign that is irrational
(Surds are turds because they don’t turd-minate) (surds are absurd numbers)
Rational - A number is described as rational if it can be written as a fraction (one integer divided by another integer). The decimal form of a rational number has either a terminating or a recurring decimal.
Irrational - A number that cannot be expressed as a fraction for any integers . Irrational numbers have decimal expansions that neither terminate nor become periodic.
Adding or subtracting radicals
- Coefficients of the radicals can be added or subtracted as long as the radicand is the same
- Different radicands cannot be combined
- Always reduce radicals to simplest form
Radicand: the number found under a radical symbol
How do you simplify radicals
Make the radicand (the number found under a radical symbol) as small as possible
1. Split the radicand into a perfect square multiplied by another integer
2. Separate into two radicals (Using a radical rule)
3. Square root the perfect square
4. Show in the form of Coefficient multiplied by radical
What are the radical rules (Give them, don’t describe)
Rule 1: √a x √a = a
Rule 2: √a x √b = √a+b
Rule 3: √a/√b = √a/b
What is a radical conjugate
If there are multiple terms in the denominator, we must use a radical conjugate to rationalize the expression.
In simple terms:
If there is more than one term in the demominator, multiply both the top and the bottom by the opposite, (+ becomes - and x becomes ÷), this allows it to still be equal, because you are multiplying it by “1”, as anything over itself is equal to one.
How do we multiply radicals
Treat radicals like a variable - Multiply the coefficients together
Remember to multiply coefficients together when using rule 1
Use radical rules 1&2 - (recall)
Rule 1: √a x √a = a
Rule 2: √a x √b = √a+b
What are the index laws
- x^m . x^n = x^m+n
- x^m/x^n = x^m-n
- (x^m)^n = x^m.n
What are the rules to simplify a radical when multiplying, and when dividing (one each)
Multiplying: √a . √b = √a.b
Dividing: √a/√b = √a/b
