Polynomial divided by monomial | Polynomial and rational functions | Algebra II | Khan Academy

TL;DR

The video explains how to simplify an expression with exponents and division using multiple approaches.

Transcript

Simplify the expression 18x to the fourth minus 3x squared plus 6x minus 4, all of that over 6x. So there's a couple ways to think about them. They're all really equivalent. You can really just view this up here as being the exact same thing as 18x to the fourth over 6x plus negative 3x squared over 6x, or you could say minus 3x squared over 6x, pl... Read More

Key Insights

• 😑 Simplifying expressions involves decomposing and distributing operations.
• 😑 Division can be applied to each term of an expression separately.
• ❓ Exponents can be simplified by subtracting 1 from the original exponent.
• ❎ Negative exponents can be expressed as fractions to eliminate negative values.
• 😑 Coefficients can be divided to simplify an expression.
• 🗂️ Dividing by a variable with an exponent of 1 results in 1.

Questions & Answers

Q: How can the expression 18x^4 - 3x^2 + 6x - 4 / 6x be simplified?

There are multiple ways to simplify the expression. One approach is to decompose the numerator and divide each term by 6x. Another method is to distribute the division and simplify the coefficients and exponents separately.

Q: What is the result after dividing 18x^4 by 6x?

Dividing 18x^4 by 6x gives 3x^3. The coefficient 18 is divided by 6, resulting in 3. The exponent x^4 is reduced to x^3 by subtracting 1 from the exponent.

Q: How can negative exponents be handled in the simplification process?

Negative exponents can be dealt with by writing them as fractions. For example, x^-1 can be expressed as 1/x. This ensures that the exponent is positive and follows the rules of exponent properties.

Q: Can the expression 4 / 6x be further simplified?

Yes, the expression 4 / 6x can be simplified. Dividing 4 by 6 gives 2/3. Multiplying this by 1/x results in 2/3x or 2/(3x). Alternatively, it can be written as -2/3x^-1, considering the negative exponent.

Summary & Key Takeaways

• The video demonstrates various methods to simplify an expression by decomposing the numerator and distributing the division.

• By dividing the coefficients and applying exponent properties, the expression can be simplified.

• The final simplified expression includes terms with reduced coefficients and exponents.