Comprehensive Algebra 2 Guide: Binomials, Exponents, and Inverses

today we're going to talk about Algebra 2 and we've got a lot to cover so let's get right to it first let's talk about

multiplying binomials a binomial may look like this x + 3 so then two binomials being multiplied together may

look like this x + 3 * x -2 notice how even though there are two binomials here they multiply two numbers together to

create one number so what would happen if we were to throw an equal Z at the end of the line x + 3 * x - 2 is equal

to 0 well then this would be two numbers that multiply together to equal 0 x + 3 as 1 and x - 2 as the other but what do

we know about multiplying two numbers together and ending up with zero well at least one of the two numbers must be

equal to zero so now both of these equations are possible answers leading to two solutions for x x = -3 and x = 2

two binomials multiplied together also assists in introducing an abbreviation that's surprisingly helpful foil or fo i

l foil stands for first Outside Inside last and refers to the four multiplications that occur when

multiplying binomials as an example x + 3 * x - 2 is x * X which is x^2 + x * -2 which is -2x + 3 * X which is 3x + 3 *

-2 which is -6 this gives us when simplified notice that this end result is a trinomial that means that in some

cases we can start with a trinomial set equal to zero and do the inverse of foiling in order to get two binomials

multiply together equal to Zer for example if x^2 + 8x + 12 is equal to 0 then we can set up a puzzle to find

which two binomials this is as follows first we place the x's in the first position since X2 is just two x's

multiply together then we fill in the other two spots by answering the following question which two numbers

that add to 8 multiply to be 12 this works this way because the term with X is 8 and the constant term is 12 the

outside and inside terms make up the term with X and the last terms make up the constant term which two numbers add

to eight that also multiply to 12 well we can do this by listing all the pairs of factors of 12 and we conclude that

the numbers are six and two and so so so x = -2 and x = -6 are the two solutions to this expression next let's

talk about rules of exponents exponents represent how many times a number is multiplied by itself so for example 5 ^

3 means 5 * 5 * 5 is equal to 125 so then any number to the power of 1 is just itself as an example 8 to the^ of 1

is equal to 8 so then important question what's 4 to the power of zero the naive answer would be zero but it's actually

one why well one is the multipli I ative identity it's the only number that you can multiply by anything to not change

its value so to multiply by no more fours is kind of like multiplying by nothing not multiplying by zero but

multiplying by nothing that is not multiplying by something better known as doing nothing at all another rule of

exponents states that a to ^ of B * a^ of C is equal to a ^ of b + C in everyday terms this means that

multiplying something by itself a set number of times and then multiplying by itself another set number number of

times is like multiplying by itself the sum of those two numbers of times for example 4 ^ 3 * 4 ^ 2 is equal to 4 ^ of

3 + 2 which is equal to 4 ^ 5 we can also use this rule to prove that if a is not equal to 0 then a ^ of 0 is equal to

1 as follows so therefore we have that what we started with is equal to what we started with multiply by 4 to the^ of 0

this would only work if 4 to the^ of 0 were Al to 1 which is the multiplicative identity now that we know about Powers

let's address square roots and radicals the square root of a number let's say 16 is defined as the number that when

multipli by itself is 16 so then what number multiply by itself is 16 yes four very good but remember something if you

multiply two negative numbers together you get a positive number so the answer is also -4 since -4 multipli by itself

is 16 we'll get into the repercussions of square roots in just a bit but there's more than just square roots if

you were to write this this would imply the square root of 16 but if you put a number in this spot this implies that

the power is different if we put a four there that means the fourth root of 16 which asks what number when multiplied

by itself four times is 16 that number as it turns out is two so the fourth root of 16 is 2 furthermore you can just

write the nth root of a number as a to^ of 1 / n this allows us to conclude things like the 3un of 10 * the 3 root

of 100 is equal to the 3 root of 10 * the 3 root of 10^ SAR which means the following so we just get 10 yay with

radicals concluded we now look at inverse operations the concept of inverse operations are that they're

operations that undo one another these have real life analog such as taking one step left and one step right when it

comes to individual operations these are relatively easy to figure out f ofx = x + 4 so the inverse of f ofx is equal to

x - 4 note that this is not F to the power of1 but rather the candidate inverse function of F so how can we do

this with more complex expressions like f ofx = 3x + 5 here's what we do since the inverse function does to X as X does

to the function then we can replace f ofx with X and the original X's with f inverse of X giving us we then isolate

the inverse function candidate as follows notice that I called it an inverse function candidate this is

because it's not necessarily the inverse function unless the function of the inverse function is equal to the inverse

function of the function of X which is equal to X another way of saying this expression is that doing the function

and then the inverse function of what you start with or doing the inverse function of the function of what you

start with must equal what you start with this comes full circle back to exponents remember how I said that

square roots include two results the positive and negative result this means that the graph depicting square roots

doesn't pass a vertical line test since there exist inputs that give more than one output this is an example where the

candidate for the square function the square root function is actually not a function at all and so it can't be the

inverse function coincidentally enough this doesn't work for any even power and it works properly for any odd power

thank you all for watching and ghf and