Find the quotient and remainder using long division. (3 x^4-5 x^3-20 x-5)/(x^2+x+3)

Name: Problem 18, Chapter 4: Algebra and Trigonometry
Uploaded: 2020-08-29T02:23:37-08:00
Duration: 2 min 5 s
Channel: Suman Saurav Thakur
Description: Find the quotient and remainder using long division. (3 x^4-5 x^3-20 x-5)/(x^2+x+3)

step 1 We have 18 number problem in which we need to divide 3x raised to the power 4 minus 5 x cube min

Find the quotient and remainder using long division. (3...

Key Concepts

Polynomial Long Division

This is a systematic method used to divide a polynomial by another polynomial of lower or equal degree, similar to the long division process with numbers. The technique involves dividing the leading term of the dividend by the leading term of the divisor, subtracting the result multiplied by the entire divisor from the dividend, and repeating the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Division Algorithm for Polynomials

The division algorithm states that any polynomial (dividend) can be expressed as the product of the divisor and a quotient plus a remainder, where the remainder has a degree that is strictly less than the degree of the divisor. This concept guarantees the existence and uniqueness of the quotient and remainder for a given division, forming the mathematical foundation for polynomial division.

Degree Considerations in Polynomial Division

When performing polynomial division, the degrees of the dividend, divisor, quotient, and remainder play a crucial role. The degree of the quotient is determined by subtracting the degree of the divisor from the degree of the dividend, and the division process continues until the degree of the remainder is lower than that of the divisor. This concept ensures that the division process terminates and the remainder is properly defined.

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