Dividing a polynomial by a monomial
Divide polynomials by monomials in Grade 9 algebra by splitting each term over the monomial denominator individually and applying the quotient rule for exponents to simplify each term.
Key Concepts
Property To divide a polynomial by a monomial, divide each term in the numerator by the denominator. $$ \frac{a+b}{c} = \frac{a}{c} + \frac{b}{c} $$ Explanation Think of the polynomial as a party size sub sandwich and the monomial as your friends. To share it fairly, you cut each section (term) of the sandwich and give a piece to every friend. This way, every part of the polynomial gets divided equally! Examples $$ (16x^3 + 8x^2 + 4x) \div 4x = \frac{16x^3}{4x} + \frac{8x^2}{4x} + \frac{4x}{4x} = 4x^2 + 2x + 1 $$ $$ (9y^4 15y^2) \div 3y^2 = \frac{9y^4}{3y^2} \frac{15y^2}{3y^2} = 3y^2 5 $$.
Common Questions
How do you divide a polynomial by a monomial?
Apply the rule (a + b)/c = a/c + b/c by separating each term of the polynomial over the monomial. Then divide each term: divide the coefficients and subtract the exponents. For (8x³ + 4x²) ÷ 4x: 8x³/4x + 4x²/4x = 2x² + x.
How do you simplify (12x⁴ - 6x³ + 3x²) / 3x²?
Separate: 12x⁴/3x² - 6x³/3x² + 3x²/3x². Divide each: 4x² - 2x + 1. Use coefficient division and subtract exponents (4-2=2, 3-2=1, 2-2=0). Final answer: 4x² - 2x + 1.
What rule governs dividing variable terms in polynomial/monomial division?
The Quotient Rule: xᵐ/xⁿ = xᵐ⁻ⁿ. Subtract the exponent of the denominator from the exponent of the numerator. If exponents are equal, the result is x⁰ = 1. Apply this rule to each term after dividing coefficients.