Question

Perform each division. $\left(x^{3}+3 x+5 x^{2}+6+x^{4}\right) \div\left(x^{2}+3\right)$

   Perform each division.
$\left(x^{3}+3 x+5 x^{2}+6+x^{4}\right) \div\left(x^{2}+3\right)$

Intermediate Algebra

Intermediate Algebra

Alan S. Tussy, R.… 5th Edition

Chapter 6, Problem 60

Instant Answer

Solved by Expert Alison Rodriguez

02/25/2023

Step 1

So, we rewrite the polynomial as $x^{4}+x^{3}+5x^{2}+3x+6$. Show more…

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Perform each division. $\left(x^{3}+3 x+5 x^{2}+6+x^{4}\right) \div\left(x^{2}+3\right)$

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Key Concepts

Polynomial Long Division

This is a method used to divide one polynomial (the dividend) by another polynomial (the divisor) similar to traditional long division with numbers. It involves writing the polynomials in descending order of degree, dividing the highest-degree terms, multiplying the divisor by the resulting term, subtracting, and repeating until the degree of the remaining polynomial (the remainder) is less than that of the divisor.

Ordering Polynomial Terms

Before performing polynomial division, it is crucial to express both the dividend and divisor in standard form, with their terms in descending order of degree. This organization allows for systematic identification of the leading term and ensures that the division process proceeds correctly.

Division Algorithm for Polynomials

This algorithm states that for any two polynomials, where the divisor is non-zero, there exist unique quotient and remainder polynomials such that the dividend equals the divisor multiplied by the quotient plus the remainder. The remainder is either zero or has a degree less than that of the divisor, paralleling the well-known division algorithm for integers.

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