State the quotient and remainder when the first polynomial...

State the quotient and remainder when the first polynomial is divided by the second. Check your division by calculating (Divisor)(Quotient) + Remainder.
$$2 x^{5}+5 x^{4}+x^{3}-7 x^{2}-13 x+12 ; \quad x^{2}+2 x+3$$

Key Concepts

Verification of Polynomial Division

After obtaining the quotient and remainder, the division can be verified by substituting them back into the relation given by the division algorithm: Dividend = (Divisor × Quotient) + Remainder. This check confirms that the division has been performed correctly and that the quotient and remainder satisfy the original polynomial equation.

Remainder

The remainder is the polynomial that remains after the division process when the divisor no longer 'fits' into the dividend. By definition, the degree of the remainder is lower than the degree of the divisor, and it represents the part of the dividend that cannot be exactly divided by the divisor.

Division Algorithm for Polynomials

The division algorithm for polynomials states that for any two polynomials, a dividend and a non-zero divisor, there exist unique quotient and remainder polynomials such that the dividend equals the divisor multiplied by the quotient plus the remainder. Furthermore, the degree of the remainder is strictly less than the degree of the divisor.

Polynomial Division

Polynomial division is a process analogous to numerical long division where a dividend polynomial is divided by a divisor polynomial to produce a quotient polynomial and possibly a remainder. This technique is used to simplify expressions and solve polynomial equations, and it is fundamental in algebra and calculus.

Quotient

The quotient in polynomial division is the polynomial that results from the division process. It represents the main part of the result when the dividend is divided by the divisor, capturing the terms that fully 'fit' into the dividend based on the degrees of the polynomials.

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