Dividing polynomials by x (no remainders) | Algebra 2 | Khan Academy

TL;DR

The video demonstrates two methods for simplifying a rational expression involving division.

Transcript

• [Instructor] What I'd like to do in this video is try to figure out what x to the fourth minus two x to the third plus five x divided by x is equal to. So pause this video and see if you can have a go at that before we work through this together. All right, so if we're saying what is this top expression divided by this bottom expression, another ... Read More

Key Insights

• 😑 Dividing a rational expression with a variable involves distributing the reciprocal to each term or factoring out the variable from the numerator.
• 😀 Dividing x^n by x^m results in x^(n-m), following the exponent rules.
• 🍃 Dividing the same variable with the same exponent cancels out and leaves only the coefficient.
• 😑 The two methods demonstrated in the video are equivalent, resulting in the same simplified expression.

Questions & Answers

Q: How can we simplify the expression (x^4 - 2x^3 + 5x) / x?

There are two methods to simplify the expression. The first method is to distribute the reciprocal of x to each term, resulting in (x^3 - 2x^2 + 5). The second method involves factoring out x from the numerator, resulting in x(x^3 - 2x^2 + 5) / x. Simplifying further cancels out the x terms, leaving (x^3 - 2x^2 + 5).

Q: What is the value of x^4 / x?

The value of x^4 / x is x^3. This is because when dividing with the same base (x), we subtract the exponents (4 - 1 = 3).

Q: How do you simplify -2x^3 / x?

To simplify -2x^3 / x, we divide the coefficients (-2 / 1 = -2) and subtract the exponents (3 - 1 = 2). Therefore, the simplified expression is -2x^2.

Q: What is the result of 5x / x?

The result of 5x / x is 5. When dividing the same variable (x) with the same exponent (1), it cancels out and leaves only the coefficient (5).

Summary & Key Takeaways

• The video discusses how to simplify the expression (x^4 - 2x^3 + 5x) / x using two different methods.

• The first method involves distributing the reciprocal of x to each term, resulting in (x^3 - 2x^2 + 5).

• The second method involves factoring out x from the numerator, resulting in x(x^3 - 2x^2 + 5) / x. Simplifying further cancels out the x terms, leaving (x^3 - 2x^2 + 5).