(a) Show that a b=(1)/(2)[(a+b)^2-(a^2+b^2)] . (b) Show that (a^2+b^2)^2-(a^2-b^2)^2=4 a^2 b^2.

step 1 Okay, hello there. We're going to be going over factoring today. And just a disclosure, this vid

(a) Show that a b=(1)/(2)[(a+b)^2-(a^2+b^2)] . (b) Show that...

Key Concepts

Sum and Product Relationships

These relationships connect the sum or difference of squares with the product of elements, allowing expressions to be rewritten in forms that bridge addition and multiplication. They are particularly useful in problems where one needs to express products (such as ab) in terms of sums of squares and squared sums, as seen in part (a) of the question.

Brahmagupta–Fibonacci Identity

This identity shows that the product of two sums of two squares can itself be expressed as a sum of two squares. It plays a critical role in decomposing expressions like (a^2 + b^2)(c^2 + d^2) into the sum of two squared terms, illustrating a deeper structure in the interplay between sums and products in algebra.

Algebraic Identities

These are foundational formulas in algebra used to transform and simplify expressions. They include expansions, such as the square of a sum, and re-grouping terms to reveal hidden factors or structures. In this question, algebraic identities are essential for rewriting products and sums in alternative forms that make factorization possible.

Difference of Squares

The difference of squares is an algebraic identity that permits expressing the difference between two squares as a product of two binomials. This concept is crucial for breaking down expressions like a^2 - b^2 or other similar structures into simpler factors, facilitating further manipulation or complete factorization.

Transcript

00:03 Okay, hello there.

00:05 We're going to be going over factoring today.

00:08 And just a disclosure, this video is for those who have already had some practice and learned some basics with factoring because these ones are a little more challenging than usual, and i won't be going over those basics.

00:19 And so this is just for those who want to get a little bit more practice with factoring, or maybe want to try a little bit more challenging exercises to get better at factoring.

00:29 Okay, so with that said, let's get started.

00:34 So our first exercise here we have 1 half times x plus y squared minus x squared plus y squared.

00:44 All right, so what i'd want to do first is i'd want to expand this out as much as i can.

00:54 And here we have a negative sign, so if this gets distributed, both x squared and y squared become negative.

01:02 So that gives us negative x squared minus y squared.

01:10 Okay.

01:12 Now i just want to focus on this section right here, not the whole equation.

01:16 So usually to denote that, i'll make this little arrow here to express that it's not equivalent to this equation here, but it's just something i want to work on or just part of it.

01:29 So now that we have a little denoting arrow.

01:36 We'll just focus on this guy here, so that'll be x plus y times x plus y.

01:45 And when we distribute, we get x squared plus x times y plus y times x, plus y times y.

01:59 All right, and then we can simplify that a little bit.

02:02 X squared plus these two are the same thing, and so when you add them together, it just doubles their value.

02:08 So plus 2xy plus y squared.

02:15 And now that we have that figured out, i'm going to get myself a little more room here.

02:21 We can plug this guy into where this guy was because it's the same.

02:28 It's the same thing.

02:29 They're equal.

02:30 So we get one half times x squared plus 2 x, y plus y squared.

02:39 And then we just finish the rest of the equation.

02:46 All right.

02:49 And so we can do a lot of simplifying within these brackets now.

02:53 So the x squared's cancel, and so do our y squared.

02:58 And that leaves us with one half times 2xy.

03:06 Our 2s cancel, and we are just left with xy.

03:11 And that's our answer.

03:15 All right.

03:16 So let's try another exercise.

03:18 I encourage you to try this on your own.

03:21 Put your video on pause.

03:24 Try it.

03:25 Once you're done, replay it and see how it matches with our answer.

03:30 Okay, so here we just distribute the x, so we get x to the fifth plus x times y to the fourth, and we get y times x to the fourth plus y to the fifth.

03:57 Sometimes when we factor and distribute, there's only one step.

04:02 There's not really any way to simplify that, so we can be done.

04:05 Sometimes it looks better when it's grouped together, but just for the sake of practicing, distributing, and factoring, here's our answer...

Please give Ace some feedback