Divide polynomials using long division

Learning Objectives Perform long division with polynomial expressions. Correctly identify the dividend, divisor, quotient, and remainder in a polynomial division problem. Express the result of polynomial division in the form P(x) = D(x)Q(x) + R(x). Properly set up division problems that include missing terms by using zero coefficients as placeholders. Determine if a binomial is a factor of a polynomial by checking if the remainder is zero. Apply the process of polynomial division to simplify rational expressions and find roots of polynomials. How can we break down a complex function like f(x) = (x^3 - 4x^2 + 2x + 1) / (x - 1) into simpler parts to analyze its behavior, especially near its asymptotes? 🤔 This tutorial covers the systematic process of polynomial long division...

TermDefinitionExample DividendThe polynomial that is being divided.In the expression (x^3 + 2x^2 - 5x - 6) ÷ (x + 1), the dividend is P(x) = x^3 + 2x^2 - 5x - 6. DivisorThe polynomial by which the dividend is divided.In the expression (x^3 + 2x^2 - 5x - 6) ÷ (x + 1), the divisor is D(x) = x + 1. QuotientThe main result of the division, representing how many times the divisor fits into the dividend.When (x^2 - 4) is divided by (x - 2), the quotient is Q(x) = x + 2. RemainderThe polynomial 'left over' after the division is complete. The degree of the remainder must be less than the degree of the divisor.When (x^2 + 3x + 5) is divided by (x + 1), the remainder is R(x) = 3. Placeholder TermA term with a coefficient of zero used to represent a missing power of the variable in a polyn...

The Division Algorithm for Polynomials P(x) = D(x) \cdot Q(x) + R(x) This rule states that any dividend P(x) can be expressed as the product of its divisor D(x) and quotient Q(x), plus its remainder R(x). This is the standard way to write the complete result of a division. Fractional Representation \frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)} This form is useful for expressing the division as a mixed rational expression, which is particularly important when finding slant asymptotes of rational functions in calculus. The Remainder Theorem If a polynomial P(x) is divided by (x - c), then the remainder is P(c). This theorem provides a quick way to find the remainder without performing long division. You can use it to check your answer. For example, the remainder of P(x)...