Math Question: i don't understand conjugates I have tried to figure it out for days I have been using the FOIL method and it just doesn't come out right

When you multiply conjugates it is similar to multiplying (a+b)(a-b) where you just get the square of the first term - the square of the second term like a^2 - b^2
Answered On 05/18/2020 16:58

Try writing in columns when you FOIL: "a^2" terms in one column "b^2" terms in a different column "ab" terms in yet another column It's a bit early, but if you had to multiply: (a+b)(2a^2+b^3)(b^4+3ab^2+5ab+1) the columns are very useful. If you're using columns you should see something like: (x+b)(x+b)= F---x*x=x^2 O---------------x*b=xb I------------------------------------b*x=bx L------------------------------------------------------b*b=b^2 but xb=bx (multiplication is commutative for numbers) so it should be three columns, not four: F---x*x=x^2 O---------------x*b=xb I----------------x*b=xb L------------------------------------------------------b*b=b^2 That's why we multiply 4 times (F,O,I,L are each a product) but get 3 terms--we can simplify by adding the two products with same degree of x and b. (If we know b=5 or whatever actual number, then we can simplify by multiplying and adding, but if it's still a variable or an unknown written with a letter like x,y,z or a,b,c, then degree matters.) So we take: F---x*x=x^2 O---------------x*b=xb I----------------x*b=xb L------------------------------------------------------b*b=b^2 which is really F---x*x=1x^2 O---------------x*b=1xb I----------------x*b=1xb L------------------------------------------------------b*b=1b^2 Though we don't usually write the "1." We add the terms in the same column--1xb+1xb=2xb--and we get: x^2+2bx+b^2 Which might be familiar if you're working on conjugates. But what if instead of: F--x^2 O----------bx I------------bx L------------------------b^2 Which gets us from 4 products to 3 terms, we could get from 4 products to 2 terms? What if instead of x^2+xb+bx+b^2= x^2+2bx+b^2 we could get down to x^2 and b^2 terms and that's it? We can, by changing F--x^2 O---------- +bx I------------ +bx L------------------------b^2 In which O,I give us +bx+bx=2bx to F--x^2 O---------- +bx I------------ -bx L------------------------ -b^2 In which O,I give us +bx -bx which =0. Combining like terms is useful, but combining like terms to cancel each other out is very useful. If we take a term (x+b) and square it we multiply 4 times--F,O,I,L--and get 3 terms from (x+b)(x+b). If we take a term (x+b) and multiply it by its conjugate (x-b) we get 4 terms that reduce to 2 terms: (x+b)(x-b)= F-----x*x=x^2 O-------x*-b = -bx I---------b*x=+bx L-------------- +b*-b = -b^2 = x^2 -bx +bx -b^2 = x^2 (-bx +bx) -b^2 = x^2 -0 -b^2 = x^2 -b^2 Of course FOIL can be used and you can't simplify by combining like terms: (a+b)(c+d)=ac+ad+bc+bd F ac O ad I bc L bd Which doesn't simplify below 4 terms. Or (x+2y)(3w+4z)=3wx+4xz+3wy+8yz. But conjugates show up from time to time. Probably the first time you see them is rationalizing denominators--you don't want a square root in the bottom of a fraction. The complex conjugate is important when working with complex numbers, and is the same basic idea. HTH.
Answered On 05/27/2020 17:10