"It's a cube minus B cube which is equal to A minus B times A squared plus AB plus B squared."
"One way to get started with mathematics is by starting with algebra, which is actually the most common way that people start learning math."
"What is easiest and most useful in algebra is such as men constantly require in cases of inheritance, legacies, partitions, lawsuits, and trade."
"Arithmetic is calculating with numbers, whereas algebra is reasoning about numbers."
"The word algebra comes from an Arabic term for reuniting broken parts."
"Complex numbers provide a visceral connection between geometry, trigonometry, and algebra."
"So even before working it out, I can know with confidence that if I were to go through the painstaking process of taking negative one half plus the square root of three over two and cubing that, I'm going to get back one."
"You don't need to know much more than algebra to do very, very well in this class."
"Inverse function: x plus 4 over 2 equals f minus 1 of x."
"What is g when you put f into it? ... 2x minus 4 in brackets squared plus 5."
"This is going to give us 3 over square root of 1 plus X to the 4th."
"Factoring comes up over and over and over again in algebra."
"You're not going to have a complete toolkit factoring toolkit if you don't understand the difference of two squares."
"The determinant of a product: determinant of A times determinant of B."
"Now our equation is looking like Y is equal to x^2 - 2x + 4."
"Once we have it, then we need to find the value of eleven a plus twelve b."
"He made it possible for algebra to exist as a subject in its own right rather than as a technique for finding numbers."
"Matrices, matrices, matrices! Matrices are an area of maths in algebra."
"Any point on the Lie algebra can be thought of as a vector from the identity. And because the Lie algebra is tangent to the Lie group, this vector is just a tangent vector of the Lie group."
"And any finite dimensional real algebra without nilpotent is copies of R, C, and H."
"Variables and constants. See, algebra isn't as useless as we all thought it was."
"Here's how to rearrange variables on the SAT: question 21 states the expression and then I'm not going to say that but you can read it it's equivalent to a x to the power B where A and B are positive constants X is greater than one what is the value of a plus b."
"The quadratic formula: negative B plus or minus the square root of B squared minus 4AC over 2A."
"Squares now mean multiplied by complex conjugate."
"Algebra, algebra made me a star. You can't be a ship's captain without passing a few math courses."
"Techniques from algebraic geometry and algebraic topology."
"Then moving through foil we've got i for inside so we do the inside we've got one times x so that just means we have one x we don't write the one we just write x."
"The derivative of x cubed is 3x squared."
"Spinners are members of minimal left ideals in Clifford algebras."
"What we can say about geometric algebra is it kind of combines align areas and volumes."
"Consider the word commutative. What do you think of when you see this word? When I look at this word, I see the word commute, which reminds me of movement, which is pretty much what the commutative property allows you to do when adding or multiplying algebraic terms."
"Now in the 19th century this was well understood, Gauss proved the fundamental theory of algebra, they knew that this should be true, they worked in the complex numbers."
"...so now we just need to combine like terms..."
"...this is what we call this in algebra this is a one variable linear equation."
"Simultaneous equations are two or more algebraic equations that share variables."
"...the algebra of infinite justice was never so rude."
"It's a miracle that the algebra which is straightforward, you really see the value of eigenvectors."
"One of the programs that we created that has turned out to be really the most effective piece of our work in all three areas is called the algebra generation."
"To get A by itself, I'm going to take away UT from both sides."
"Are there people that use algebra in real life? I don't know."
"Completing the square is going to tie directly into the next topic we're going to talk about, which is the very famous quadratic formula."
"The reason you're learning completing the square is because up until now in algebra you only know how to solve very special quadratic equations."
"If you're any sort of algebra student... you're gonna have to have a real command of what the domain and range is."
"But in the end, what that tells us is that what we either have f of x is always equal to x or f of x is always equal to negative x."
"Whenever you subtract polynomials, you have to distribute the negative sign."
"I said it's like trying to teach a monkey how to do [__] algebra."
"A group has a multiplication, it has an identity element, and it has an inverse for each element."
"All non-zero members of the integers mod prime have multiplicative inverses, guaranteeing that the integers mod prime will always form a field."
"The integers mod p, where p is not a prime number, do not form a field because some elements don't have multiplicative inverses."
"Quaternions are kind of cool. They have length, you can add them, you can subtract them, you can multiply them, you can divide them. Everything about them is really cool."
"Each quaternion has an inverse. That means I can divide. Because I can take one quaternion Q one divided by the other one Q two, all I have to do is take Q one times Q two inverse."
"These things are kind of cool. They're just like numbers. Unfortunately, they don't have this commutative property. You have to watch out when you multiply because a times B is not equal to B times a. That's what's wrong with quaternions in the world."
"What this quaternion represents is a rotation of angle theta in the positive cupping direction of my fingers around the vector V. A rotation in the cupping direction of my fingers around the vector V."
"Algebra is really more of a language of modern mathematics."
"The equation of a straight line is given as y equals MX plus C."
"Collecting like terms: terms with the exact same variables and exponents are like terms."
"Use the distributive property to multiply a monomial by a polynomial."
"Add or subtract polynomials by collecting like terms."
"Solving equations: isolate the variable by moving all other numbers away from it."
"Add or subtract the same thing to both sides of the equation to keep it balanced."
"Remember, you can plug your answer back into the equation to check your work."
"G is equal to 2 and you can check your answer by plugging it back into your equation. Is 2 minus 5 equal to negative 3? Yes, so we have the right answer here."
"Isolate the U that is currently being multiplied by five. We do the opposite of multiplying by five, which is dividing by five."
"First thing you want to do is you want to isolate the term that has the variable."
"Now the Y is being multiplied by 7. To separate a coefficient from a variable like this, we have to divide both sides by the 7."
"What you want to start by getting both of those terms with the variable on the same side of the equation."
"If you have fraction equals fraction, you can use what's called cross multiplication."
"If we want to get rid of the brackets, we can multiply both sides by 30."
"If you have more than one fraction, you can get rid of both of them at the same time by multiplying both equations by a common denominator."
"The standard equation of a circle is x minus h squared plus y minus k squared equals the radius squared."
"If we can factor, we can use a zero product property."
"F of x equals (x + 6)(x - 2)(x - 4)."
"Everything you need to Ace pre-algebra and Algebra 1 in one big fat notebook."
"It's not a perfect book because it doesn't have as much knowledge as say you know college algebra by Blitzer but it's got answers to everything which make it a great book to start doing math as soon as you get it."
"I think that you basically will master algebra and be able to solve any type of problems."
"So, because we've got a b in the denominator and a b in the numerator, those b's cancel."
"You can learn Calculus if you know a little bit of algebra."
"The distributive property of multiplication says that a times b plus c is going to be equal to a times b plus a times c."
"The commutative property of multiplication says you can multiply in any order and get the same exact result."
"All like terms have matching variables: x and x, x2 and x2, xy and xy, a2b and a2b."
"With fractional exponents, the denominator is the root, the numerator is the power."
"...variables where we'd expect numbers and numbers where we'd expect variables but remember that algebra is your friend."
"Remember what we covered at the beginning about simultaneous equations. The GMAT loves to give you simple algebra and a bunch of simple steps together end up seeming complex."
"Remember that you don't want to be testing cases, you want to be thinking logically. Algebra's your friend, guessing numbers testing cases is not your friend."
"When you see exponents with the same base an addition or subtraction, immediately think: what's the biggest thing I could factor out from everything?"
"A polynomial is a function \( f(x) = a_n x^n \) all the way down to \( a_1 x + a_0 \)."
"The highest nonzero power of \( n \) is called the degree of \( F \)."
"A complex number is said to be algebraic over the rationals if it is the root of some polynomial with integer coefficients."
"Every polynomial over the integers completely factors, we say splits, over \( \mathbb{C} \)." - This is known as the fundamental theorem of algebra.
"You're basically making discoveries and solving algebraic equations by playing games."
"A ring is a set R together with two operations: addition and multiplication."
"For all little r in R, we have r cubed equals r, then R is commutative."
"If R S equals S R for all R and S in R, then we say that R is commutative."
"Six divides n times n plus 1 times n plus 2 for any integer n."
"The axis of symmetry is simply x equals negative one."
"For any cubic function, domain and range is just real numbers."
"The eigenvalues of the Hecke operators are algebraic integers."
"That sequence there has an nth term n plus 13."
"To find a turning point, we have to do something called completing the square."
"If we expand this bracket... we would get x squared plus eight x but we would get plus sixteen at the end."
"The coordinates of our turning point are negative four and negative thirteen for the y coordinate."
"When you get this sort of scenario here, you get two solutions because there's two crossover points."
"When a quadratic doesn't factorize, you've got to use the quadratic formula."
"All we're doing is we're rewriting it in a different form, in its completed square form."
"Now we can go back, actually get this into a single line equation."
"That's going to allow me to get zero over here."
"So what is G when you put F into it?"
"The equation of the circle was x squared plus y squared equals 29."
"In algebra, when multiplying, we can add the powers."
"Completing the square is absolutely essential if you're studying algebra or beyond."
"Stick with me on this journey, I guarantee you that you'll be comfortable in algebra but not only that you'll be really really good."
"Plan for algebraic properties and compositionality, and don't always just reach for the most expressive thing that you could."
"That is the power of refactoring; that's the power of algebra."
"Algebraic data types are insanely powerful; they're so powerful that you can create a precise data model of almost anything you can imagine."
"Let's move that 3x over; I'm going to move the 3 over as well."
"And there's a quadratic in equals 0 form."
"It's important to realize that this actually represents two different numbers."
"If it were n's, it would factor to (n - 2)(n + 1)."
"Here the only thing that's different is that it's e to the x's."
"So the solutions are either where the first bracket is 0 or the second bracket is zero."
"The discriminant is just the part under the square root in the quadratic formula."
"The fundamental theorem of algebra says there will always exist a solution."
"Alpha is the greatest root of the cubic polynomial x cubed minus three x squared plus one equals zero."
"When you have a binomial raised to the fourth power, immediately think of the difference of two squares."
"When you have a trinomial, you're automatically going to be thinking factoring."
"Whenever you have a binomial multiplied by its conjugate, it's going to produce your Pythagorean identities."
"Y equals two-thirds X plus three."
"We can make algebra truly real for all students and at the same time, we get computer science for all."
"So, at that point, four by x is four x minus sixteen."
"Slope-intercept form is y equals mx plus b, where m is slope and b is your y-intercept."
"The solution to the system is that intersection point of the two lines."
"Algebra is the study of structures, symmetry, and patterns. It's all about identifying patterns."
"In many ways, I would say that algebra is the purest form of mathematics."
"When the bases are the same, you can start adding or subtracting exponents."
"When we do the algebra, we find that the velocity when it's at point B is the square root of 3GL."
"Y squared is equal to 14 squared plus 6 squared."
"To know the gradient, we want to put it in the form y equals mx plus c."
"Y equals MX plus B is probably one of the most important equations in all of algebra."
"You will have a very good feel for what algebra is about."
"Variables are nothing more than placeholders."
"Everything you learn back in elementary school, you're going to follow those same rules of arithmetic with algebra."
"We're really getting heavy-duty into the use of variables in algebra."
"The graphical interpretation of an algebraic expression is very important."
"An equation is almost kind of like a riddle or a challenge."
"We're always looking for solutions in algebra."
"Inequalities are another topic that we study in algebra."
"Being able to graph lines is a huge part of algebra."
"Algebra 1 is really not much more than variables, expressions, graphs, and rules."
"Y equals AX has lots and lots of applications."
"We need to talk very seriously about something called rationalizing the denominator."
"Every time you multiply by a conjugate, you will kill square roots every single time."
"The only reason that we have no radicals in the answer is because of this beautiful cancellation in the middle."
"This guy here, this 3 minus the square root of 7, is what we call the conjugate."
"If we multiply by a very slightly modified copy of ourselves, where we only do one thing that's changed the sign, we eliminate all radicals in the answer."
"When you multiply a binomial with the radical like this times its conjugate, we will always have no radicals in the answer."
"That process is what we call rationalizing the denominator."
"When you multiply by a conjugate, you kill all the radicals."
"In order to rationalize the denominator of any large fraction like that, we're going to multiply the top and the bottom by what we call the conjugate."
"The idea of a conjugate is not going to go away, it's just a special way of multiplying by a very similar term that gets rid of the radicals."
"If you have a squared minus B squared, which is the difference of two squares, it's always written as A plus B times A minus B."
"This radical up here should not scare you."
"These hundreds of pages of algebra collapse to a single term."
"Scipione del Ferro, a mathematics lecturer at the University of Bologna, found a general method of solving cubic equations."
"The ARS Magna became one of the most important algebra books of all time."
"Algebra changed dramatically throughout the 19th century; in 1800 the subject was about solving equations, but by 1900 it had become the study of mathematical structures."
"It was the great Carl Friedrich Gauss who showed that they do; this is called the fundamental theorem of algebra."
"The literal translation of the word algebra is to restore and to rebalance."
"Because the word quadratic means squared, the formula is going to be y equals a with x squared."
"The a value is also called the vertical stretch factor because the a value tells you how thin or how wide the graph is going to be."
"If you truly want to learn algebra with me as your teacher, then you want to check out my pre-algebra and algebra one courses."
"So many students struggle with this very critical algebra skill."
"Linear growth... this is a huge topic in algebra."
"This type of problem is a problem that you need to understand if you want to be successful in algebra."
"If something in the parenthesis is squared, that means you include the negative."
"This is going to be applicable to any of you out there that are studying algebra."
"We are talking about equations, and in algebra, there's different type of equations."
"This is technically what we call partial fraction decomposition."
"The whole entire goal of partial fraction decomposition is to find out what A and B are equal to."
"A is equal to 2 and B is equal to 1."
"Notice I didn't say don't do algebra, because if you know how to solve something algebraically and you feel really confident about it, you can go ahead and do that."
"Algebra is a great tool; it just makes solving problems like this so much easier."
"Don't be afraid of it, and you're going to see how easy it is to solve this problem using algebra."
"Solving advanced polynomial equations is a core skill set in algebra."
"A scalar is a single number, a vector is a one-dimensional list of numbers, and a matrix is a two-dimensional grid of numbers."
"When you're using the algebra side of your brain, you don't completely shut off the logic side of your brain, and when you're using the logic side of your brain, you don't completely shut off the algebra side of your brain."
"We want to get the variable by itself on one side of the equation."
"A to the minus 1 plus 'b' to the minus 1 is not the same thing as 'a' plus 'b' to the minus 1."
"First rule of thumb is get rid of those negative powers, rewrite them so that they're positive powers."
"Always rewrite negative exponents immediately."
"A lot of algebra is just arithmetic but with letters."
"When we start solving equations and inequalities, factoring is a huge benefit to us."
"Algebra is simply a tool, and the main concept of algebra is that we use variables, things like X, Y, and Z, to represent unknown values."
"Lo and behold, you get 1X and 2X which adds to 3X, so there's a nice factorization."
"Difference of squares, right. X squared minus 1, you say, but 1 is 1 squared, so it's a difference of squares."
"To find the inverse of a function, every X swaps with Y and every Y swaps with X."
"An algebra starts with a vector space and then endows it with a multiplication."
"The quadratic formula reads: opposite B plus or minus the square root of B squared minus 4AC all over 2A."
"You want to get the variables on one side and the numbers on the opposite side."