Polynomial Division
Polynomial division is a Grade 7 advanced math skill from Yoshiwara Intermediate Algebra covering long division and synthetic division of polynomials. Students divide a polynomial by a linear or polynomial divisor and express the result as quotient plus remainder over divisor.
Key Concepts
Property An algebraic fraction is 'improper' if the degree of the numerator is greater than or equal to the degree of the denominator. We can simplify it by division. The result is the sum of a polynomial and a simpler fraction.
If the divisor is a monomial, divide it into each term of the numerator: $$\frac{9x^3 6x^2 + 4}{3x} = \frac{9x^3}{3x} \frac{6x^2}{3x} + \frac{4}{3x} = 3x^2 2x + \frac{4}{3x}$$.
If the divisor is a polynomial, use a method similar to long division. The final answer is expressed as: $$\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$$.
Common Questions
How do you perform polynomial long division?
Divide the leading term of the dividend by the leading term of the divisor. Multiply, subtract, bring down the next term, and repeat until the degree of the remainder is less than the divisor.
What is synthetic division?
Synthetic division is a shortcut for dividing a polynomial by a linear factor (x - c). It uses only coefficients and is faster than long division.
How do you write the result of polynomial division?
The result is written as: quotient + remainder/divisor. For example, (x^2 + 3x + 2) ÷ (x + 1) = x + 2.
What does the remainder theorem say about polynomial division?
When polynomial p(x) is divided by (x - c), the remainder equals p(c). This lets you evaluate polynomials quickly.