معرفی جبرفصل: بخش ریاضی / درس: جبر، معادلات و نابرابری ها / درس 1
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Intro to Algebra
Introduction to Algebra. And we’ll be starting our discussion of algebra with algebraic expressions. So let’s start at the very beginning. First of all, just what is algebra? Well, as it turns out, the world actually has a sophisticated definition of mathematics, overall.
But that doesn’t concern us. For our purposes, algebra is the part of math that involves variables. So as soon as variables show up in math problems, that’s algebra. Let’s define some terms. A variable is a letter that represents either a specific number or all numbers. And this is a subtle distinction here.
In algebraic expressions, the variable represents either all numbers or all numbers with very few exceptions, and the point in algebraic expressions is to find or use patterns true for all numbers. In algebraic equations, by contrast, the variables represent, one or two initially unknown variable values. And the point is to solve for those specific values.
So, with equations we’re trying to solve. With expressions we’re just trying to find patterns that are true for all numbers, with the variables representing all numbers. So, this is a subtle distinction. And we’ll talk about this more. So more terms.
A constant is a number, or a symbol, such as pi, that doesn’t change in value. So, just constant is just a fancy word we use for ordinary numbers. A term is a product of constants and variables, including powers of variables. So for example, all of these are terms. It could be just a single number by itself, a single variable by itself, or numbers times variables and that includes powers of variables.
The coefficient is the constant factor of a term. So for example, 6y squared, the variable y squared is being multiplied by the number 6. So 6 is the coefficient. Now if we look at x, just x by itself.
You might think well gee, that has no coefficient. And this is subtle, when no coefficient is written, the coefficient is one. Because of course X equals one times X. We wouldn’t write the one. That would be kind of redundant, but the point is X has a coefficient of one. And in fact that last one x to the 5th, y to the 6th, z to the 7th.
That last term, also has a coefficient of 1. Now these three terms, constant, term and coefficient, are terms that could show up. On the test. The, the test question might actually mention one of these in asking the question. So these are very important to know.
All the other terms that will come in this video are terms that we’ll need in our discussion of algebra, but will not be things you’ll need to know for the test. An expression is a collection of one or more terms joined by addition or subtraction. So all of these, are examples of algebraic expressions. Notice, expressions don’t have equal signs.
So once we get into the business of taking an expression and setting it equal to a number, and solving, well then we’re talking about algebraic equations. We’ll get there, but that’s not what these first few videos are about. Here, we’re talking about individual expressions. Now we may say that one expression is always equal to another expression, but the point is, each expression by itself is a thing without an equal sign.
That’s very important to realize. A monomial is an expression with exactly one term. All of these are monomials. They are single terms. It could be just a number, could be just a variable. Or a product of numbers and variables.
A binomial is an expression with exactly two terms. Here, a bunch of binomials. Some of them simple. Some of them not so simple. A trinomial is an expression with exactly three terms. And those are some examples of trinomials.
Now we don’t have to worry about, what if there are four terms, what if there are five terms. We don’t have to worry about those, the words for those. A polynomial is an expression with any number of terms, involving only one variable. So for example, this one here is a polynomial because the only variable is x.
This one involves more than one variable. So technically, that’s not a polynomial. A linear term, is a term with a single power of a variable, so there’s no exponent written. So, just when you have a variable by itself or a number times a variable, that’s a linear term.
A quadratic is a term with a squared variable. X squared or y squared. A cubic term is a term with a cube of a single variable, y cubed or x cubed. And we will not need terms for higher degrees. Now we’ll make a distinction, and this is a little tricky. The words linear and quadratic can describe individual terms, but they can also describe the entire expression, involving a single variable.
In a linear expression, the highest power of the variable is 1. So, 17x is a linear monomial. 3x minus 5 is a linear binomial. So one of them, it’s a one term thing, the other is a two term thing. And notice we don’t have many options here. A linear binomial must have one linear term and one constant term.
As long as we’re dealing with only one variable, those are the only two options for the two terms. It’s a little more interesting when we talk about quadratics. In a quadratic expression the highest power of the variable is 2. So 14x squared, that is a quadratic monomial.
A single term, where the highest power is 2. X squared minus 4 is quadratic binomial with a quadratic term, x squared, and a constant term, negative 4. 2x squared plus 8x is a quadratic binomial with a quadratic term, 2x squared, and linear term, 8x. X squared minus 10x plus 25 is a quadratic trinomial with a quadratic term, the x squared, a linear term, negative 10x, and a constant term.
And so for example we could ask, what is the coefficient of the linear term? The coefficient of the linear term here, is negative 10. Many of the upcoming videos concern quadratic trinomials. That’s a very, very big topic in algebraic expressions. We have a bunch of videos on that topic. In the study of algebraic expressions, we are looking for rules or patterns true for all numbers.
In the study of algebraic equations, our goal is to solve, so again, these first few videos are just about algebraic expressions, then we’ll get to algebraic equations. We discussed several terms, variable, constant, term, coefficient, expression, monomial, bino, binomial, trinomial, polynomial, linear, quadratic, and cubic. And these are terms important to know because we will be using these terms in our discussion of algebra.
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