Chapter 6.7 - Dividing Polynomials

What are the 2 Steps When Dividing a Polynomial by a Monomial? (a + b) ÷ c

1) Divide each term of the Polynomial by the Monomial (based on Distributive property) 2) Combine quotients with correct signs = (a + b) _______ c a b = − + − c c

Remember this RULE: When an EXPONENT is ONE, what is the answer? Example: (x + y)¹ = ?

Always the same as the number/variable/term... Same with: 8¹ = 8 x¹ = x (4x²y)¹ = (4x²y)

Remember this RULE: When an EXPONENT is ZERO, is what? Example: (x + y)⁰ = ?

Answer is always 1 Same with: (4x²y)⁰ = 1

Closure of Integers & Polynomials

Means that when you ADD, SUBTRACT, OR MULTIPLE you Always get another Integer. 2 + 3 = 5 or 3 - 2 = 1 or 2 x 3 = 6 Same is true with POLYNOMIALs a + b or a - b or a b

How do you CHECK your work?

Multiply the DIVISOR (what you divided with) by your ANSWER. Look at this example closely.

Are Integers and Polynomials CLOSED under DIVISION?

NO = 3/4 will NOT give you an INTEGER x² = − = x⁻⁴ <--- NO CLOSURE x⁶ BECAUSE: x⁻⁴ is NOT a Positive Exponent and Polynomials MUST have Positive Exponents

Sometimes Division has a POSITIVE Exponents x⁵ − = ? x³

SUBTRACT EXPONENTS (numerator - denominator) x⁵ = − x³ = x²

Sometime Division has NEGATIVE Exponents x⁵ ÷ x⁸ = ?

SUBTRACT EXPONENTS & CONVERT NEGATIVE TO POSITIVE (by moving negative exponent to denominator and making positive) = x⁵ ÷ x⁸ = x⁻³ OR 1 = − x³

Sometime Division has POSITIVE and NEGATIVE Exponents = 8x⁴y² ÷ 16xy⁶

Xs are POSITIVE EXPONENTS & Ys are NEGATIVE = 8x⁴y² ÷ 16xy⁶ = 1x⁴y² ÷ 2xy⁶ = (1x⁴⁻¹) ÷ (2y⁶⁻²) = 1x³ ÷ 2y⁴