Polynomial long division
Perform polynomial long division by dividing term by term: match leading terms, multiply, subtract, bring down, and repeat until the remainder has lower degree than the divisor.
Key Concepts
Property Polynomial long division is a method for dividing a polynomial by another polynomial of the same or lower degree. The final result is expressed in the form: $$\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$$.
To divide $(x^2 + 7x + 10)$ by $(x + 2)$, we perform long division to find the quotient is $(x + 5)$ with a remainder of $0$. To divide $(2x^3 3x^2 10x + 3)$ by $(x 3)$, the result is a quotient of $(2x^2 + 3x 1)$ with a remainder of $0$. When dividing $(x^2 + 5x + 8)$ by $(x + 3)$, the quotient is $(x + 2)$ with a remainder of $2$, written as $(x+2) + \frac{2}{x+3}$.
Think of this as the algebraic version of dividing 125 by 5. You divide the leading terms, multiply the result by the divisor, subtract the product, and bring down the next term. This process repeats until you have a remainder that is of a lower degree than the divisor, systematically simplifying complex polynomial fractions.
Common Questions
What are the steps for polynomial long division?
Divide the leading term of the dividend by the leading term of the divisor to get the first quotient term. Multiply the entire divisor by this term. Subtract from the dividend. Bring down the next term. Repeat until the remainder's degree is less than the divisor's degree.
How do you handle a missing degree in the dividend during polynomial long division?
Insert a zero coefficient for any missing degree before starting division. For example, dividing x^3+5 by x-2, write the dividend as x^3+0x^2+0x+5 so that every degree has a placeholder column aligned correctly throughout the process.
What does the remainder mean in polynomial long division?
The remainder is the leftover polynomial after full division. If the remainder is zero, the divisor is a factor. By the Remainder Theorem, dividing p(x) by (x-a) gives remainder p(a), which equals the value of the polynomial at x=a.