Long Division With Polynomials - The Easy Way!

Summary

In this video, the process of dividing polynomials using long division is explained in detail with multiple examples. The video walks through each step of the division process: dividing, multiplying, and subtracting. Key examples include dividing expressions like x^2 + 5x + 6 by x + 2, 2x^3 + 8x^2 - 6x + 10 by x - 2, and 6x^4 - 9x^2 + 18 by x - 3. The step-by-step instructions and visual aids help viewers understand polynomial long division, including handling remainders, ensuring clarity at every stage of the process.

Takeaways

• 😀 Polynomial division involves dividing the terms of the dividend by the divisor, step by step.
• 😀 Always start by dividing the leading term of the dividend by the leading term of the divisor.
• 😀 After division, multiply the divisor by the result and subtract from the dividend.
• 😀 The remainder is the part left after performing the division; it can be expressed as a fraction if non-zero.
• 😀 Each division step involves dividing, multiplying, and subtracting to simplify the polynomial.
• 😀 If the remainder is zero, the division is exact, and the quotient is the solution.
• 😀 Dividing by a linear binomial (like x + 2 or x - 2) follows a consistent process of dividing, multiplying, and subtracting.
• 😀 When dividing polynomials of higher degrees, ensure to include any missing powers of x as 0x terms.
• 😀 The division process is repeated for each term in the dividend, ensuring all terms are simplified.
• 😀 In case of a remainder, express it as a fraction added to the quotient, i.e., 'remainder / divisor'.
• 😀 Polynomial long division is a reliable method for simplifying complex expressions and finding solutions to polynomial equations.

Q & A

What is the first step in dividing polynomials using long division?

The first step is to divide the leading term of the numerator by the leading term of the denominator.

Why do we subtract the exponents when dividing terms with variables?

We subtract the exponents when dividing terms with variables because we are following the laws of exponents, which state that when dividing powers with the same base, you subtract the exponents.

What do we do after dividing the terms in polynomial long division?

After dividing the terms, the next step is to multiply the divisor by the quotient we just found, then subtract the result from the current terms of the polynomial.

In the first example, what is the quotient when dividing x^2 + 5x + 6 by x + 2?

The quotient is x + 3, and there is no remainder.

What is the remainder when dividing 2x^2 + 8x^2 - 6x + 10 by x - 2?

The remainder is 46.

How do we handle a situation where there is a remainder in polynomial division?

When there is a remainder, we write the final answer as the quotient plus the remainder divided by the divisor.

In the second example, what do we get after dividing 2x^3 + 8x^2 - 6x + 10 by x - 2?

The quotient is 2x^2 + 12x + 18, with a remainder of 46.

How do we subtract terms during polynomial long division?

We subtract terms by changing the sign of the second term and then combining like terms.

What is the significance of the 'bring down' step in long division?

The 'bring down' step refers to bringing down the next term of the dividend after subtracting the multiplied result, so we can continue dividing the remaining terms.

In the third example, what is the remainder when dividing 6x^4 - 9x^2 + 18 by x - 3?

The remainder is 453.