Operations with polynomials — Basic example | Math | SAT | Khan Academy

Summary

In the transcript, the instructor simplifies a complex mathematical expression involving powers of y. Starting with 1/88 or one over 88, the expression is expanded to include y to the 100th power plus one, then subtracting 1/44 y to the 100th minus 1/2. By distributing the negative sign and combining like terms, the instructor simplifies the expression to negative one-eighth y to the 100th plus three halves, demonstrating a clear step-by-step process to reach the simplified form.

Takeaways

• 🔢 The original expression is equivalent to \( \frac{1}{88} \) or \( \frac{1}{88}y^{100} + 1 \).
• 📝 The expression is rewritten to subtract \( \frac{1}{44}y^{100} - \frac{1}{2} \) from \( \frac{1}{88}y \).
• 🔍 Negative signs are distributed to simplify the expression.
• ➖ The terms \( \frac{1}{88}y^{100} \) and \( -\frac{1}{44}y^{100} \) are combined.
• ➕ The expression \( -\frac{1}{2} \) is converted to \( +\frac{1}{2} \) after distribution.
• 🧩 The combined terms are simplified to \( \frac{1}{88}y^{100} - \frac{1}{44}y^{100} \) and \( +\frac{3}{2} \).
• 📉 The common denominator of \( \frac{1}{88} \) and \( \frac{1}{44} \) is found to be 88.
• 🔄 The fraction \( \frac{1}{44} \) is converted to \( \frac{2}{88} \) to match the common denominator.
• 📌 The simplified expression becomes \( -\frac{1}{88}y^{100} + \frac{3}{2} \).
• 🔚 The final simplified form of the expression is \( -\frac{1}{88}y^{100} + \frac{3}{2} \).

Q & A

What is the original expression the instructor is trying to simplify?

The original expression is 1/88 y to the 100th power plus 1, minus (1/44 y to the 100th minus 1/2).

What is the first step the instructor takes to simplify the expression?

The first step is to rewrite the expression and distribute the negative sign across the terms within the parentheses.

How does the instructor represent the negative distribution of the terms?

The instructor represents it as -1 times each term inside the parentheses, which results in -1/44 y to the 100th and +1/2.

What is the next step after distributing the negative sign?

The next step is to combine like terms, specifically the terms involving y to the 100th power.

What is the common denominator used to combine the terms involving y to the 100th power?

The common denominator used is 88, which is the least common multiple of 44 and 88.

How does the instructor simplify 1/88 minus 1/44?

The instructor simplifies it by recognizing that 2/88 is equivalent to 1/44, and thus 1/88 minus 2/88 equals -1/88.

What is the final simplified form of the original expression according to the instructor?

The final simplified form is -1/88 y to the 100th plus 3/2.

What mathematical property does the instructor use to combine the fractions 1/88 and 1/44?

The instructor uses the property of finding a common denominator to combine the fractions.

How does the instructor handle the term 'plus 1/2' after distributing the negative sign?

The instructor correctly keeps the 'plus 1/2' term as it is, as it is not affected by the negative distribution.

What is the significance of the term 'plus 3/2' in the final simplified expression?

The term 'plus 3/2' is the result of adding 1 and 1/2, which simplifies to 3/2, and it is a constant term in the expression.

Can the expression be further simplified after the instructor's final step?

No, the expression has been simplified to its most basic form, with the variable term and the constant term separated.

Outlines

00:00

📚 Simplifying Complex Fractions and Expressions

In this paragraph, the instructor is simplifying a complex mathematical expression. The original expression is equivalent to '1/88' or '1/88 y to the 100 plus 1, minus 1/44 y to the 100 minus 1/2'. The instructor rewrites the expression, distributing the negative sign to each term within the parentheses. This results in '1/88 y to the 100 - 1/44 y to the 100 + 1/2'. The next step is to combine like terms, specifically the fractions with 'y to the 100'. The common denominator for 1/88 and 1/44 is 88, and after combining these terms, the result is '-1/88 y to the 100'. The final simplified expression is '-1/88 y to the 100 + 3/2', which is the sum of the simplified fraction and the constant term.

Keywords

💡Equivalent

Equivalent in this context refers to having the same value or meaning as something else. It is a mathematical term used to compare expressions or equations to determine if they yield the same result. In the video, the instructor is trying to find an expression that is equivalent to a given complex mathematical expression, showing how different parts of the equation can be manipulated to achieve this equivalence.

💡Distribute

Distribute is a mathematical operation where you multiply each term inside a set of parentheses by a number outside the parentheses. It is a fundamental concept in algebra used to simplify expressions. In the script, the instructor uses distribution to simplify the expression by multiplying each term inside the parentheses by -1, which helps in breaking down the complex expression into simpler parts.

💡Negative Sign

A negative sign in mathematics is a symbol (-) that indicates the subtraction of a number or the presence of a negative value. In the video, the instructor uses the negative sign to indicate subtraction, particularly when distributing the negative sign across the terms in the expression, which changes the signs of the terms being subtracted.

💡Least Common Multiple (LCM)

The Least Common Multiple is the smallest number that is a multiple of two or more numbers. It is used in mathematics to find a common denominator for fractions. In the video, the instructor identifies 88 as the LCM of 44 and 88, which allows for the simplification of fractions by converting them to have a common denominator, facilitating the addition or subtraction of fractions.

💡Distribute

Distribute in the context of the video refers to the process of applying a mathematical operation (like multiplication or addition) to each term within a set of parentheses. The instructor uses this concept to simplify the expression by distributing the negative sign, which involves multiplying each term inside the parentheses by -1.

💡Addition

Addition is one of the four basic operations in arithmetic, where numbers are combined to find their total or sum. In the video, addition is used to combine terms after they have been simplified, such as adding 1/2 to another term, which is a key step in simplifying the overall expression.

💡Subtraction

Subtraction is the arithmetic operation of taking one number away from another. It is used in the video to simplify the expression by subtracting fractions or terms, such as subtracting 1/44 from 1/88, which helps in reducing the complexity of the equation.

💡Fractions

Fractions represent a part of a whole, expressed as a ratio of two integers. In the video, fractions are used extensively in the mathematical expressions, and the instructor demonstrates how to manipulate and simplify them, such as finding a common denominator or adding fractions.

💡Exponents

Exponents are used in mathematics to denote repeated multiplication of a number by itself. In the script, the term 'y to the 100th power' is an example of an exponent, where y is multiplied by itself 100 times. This concept is crucial in understanding the complexity of the expressions being simplified in the video.

💡Simplify

Simplify in mathematics means to make an expression or equation easier to understand or work with by reducing it to its most basic form. The instructor in the video is focused on simplifying a complex mathematical expression by using various algebraic techniques, such as combining like terms and finding common denominators.

💡Common Denominator

A common denominator is a number that is a multiple of the denominators of two or more fractions, allowing those fractions to be added or subtracted. In the video, the instructor identifies 88 as the common denominator for 1/88 and 1/44, which is essential for combining these fractions into a single term.

Highlights

The problem involves simplifying an expression equivalent to 1/88 or 1/88.

The expression includes y to the 100th power plus one, and subtracting 1/44 y to the 100 minus 1/2.

Rewriting the expression to distribute the negative sign and simplify.

Simplifying by combining terms involving y to the 100th power.

The terms 1/88 y to the 100 and -1/44 y to the 100 are combined.

The expression simplifies to (1/88 - 1/44) y to the 100 plus 3/2.

Finding a common denominator of 88 for the fractions.

Converting 1/44 to 2/88 to have a common denominator.

Subtracting the fractions 1/88 - 2/88 to get -1/88.

The final simplified expression is -1/88 y to the 100 plus 3/2.

The least common multiple of 44 and 88 is 88, which is used to combine fractions.

The instructor emphasizes the importance of distributing the negative sign correctly.

The process demonstrates the steps to simplify complex algebraic expressions.

The solution involves combining like terms and simplifying fractions.

The final result is presented clearly, showing the simplified form of the expression.

The instructor provides a step-by-step approach to simplifying the given expression.

The method used is applicable to similar algebraic simplification problems.

Transcripts

- [Instructor] Which of the following is equivalent

to all of this stuff right over here.

One over 88, or 1/88, y to the 100 plus one,

minus this entire expression,

1/44 y to the 100 minus 1/2.

So let me just rewrite it.

So this part right over here,

this is going to just be one over 88 y,

let me make sure it looks clear,

y to the 100th power plus one,

and then we're gonna subtract all of this.

So one way to think about it is we could distribute

this negative sign, so it's gonna be,

it's going to be negative,

one way you could think about it is negative one,

negative one times all of this business.

So it's gonna be minus one over 44 y to the 100th,

and then a negative, remember, we're gonna distribute it,

so negative one times negative 1/2

is going to be plus, plus 1/2.

Now, let's see what we can do to further simplify it.

So this is 1/88 y to the 100.

This is minus, or you could say negative,

1/44 y to the 100th.

And so we can add these two terms together.

This is going to be, we could write this,

one over 88 minus one over 44

y to the 100th,

and then you have plus one, plus 1/2,

so that's pretty straightforward.

So you have plus one, plus 1/2,

so that's gonna be plus 3/2, plus 3/2.

And now we just have to figure out what this is.

So 1/88 minus 1/44, well, the common denominator here

could be 88.

That is the least common multiple of 44 and 88.

88 is divisible by 44.

So if we multiply the denominator by two, you get 88.

You multiply the numerator by two, you get two.

2/88 is the same thing as 1/44.

So this 1/88 minus 2/88.

Well, this is going to give us, this right over here

is going to give us, if we,

this is going to give us minus, or negative, 1/88.

1/88 minus 2/88 is going to be negative 1/88,

y to the 100, y to the 100,

plus 3/2, plus 3/2.

And that is negative 1/88th y to the 100th,

y to the 100th plus 3/2.

This choice right over here.