2. Long DivisionWe can divide polynomials using steps that are similar to the steps of numerical long division a Notation: a ÷ b = = b a b Vocabulary: dividend ÷ divisor = quotient
3. Example: Numerical Long DivisionDivide using long division. (Set up, Divide, Multiply, Subtract, Bring Down, Repeat) 672 ÷ 21
4. Polynomial Long DivisionDividing polynomials is useful when we are trying to factor polynomials, especially when we are unsure of factors.
5. The Division Algorithm for PolynomialsAn algorithm is a specific set of instructions used to solve a problem.The Division Algorithm for Polynomials is a generalized version of the technique of long division in arithmetic. To divide polynomials, list polynomials in standard form with zero coefficients where appropriate.
6. The Division Algorithm for PolynomialsYou can divide a polynomial, P(x), by a polynomial, D(x), to get a polynomial quotient, Q(x) and a polynomial remainder, R(x). Set up, Divide, Multiply, Subtract (change signs), Bring Down, Repeat Q( x) D( x) P( x) O R( x) The process stops when the degree of R(x) is less than the degree of the divisor, D(x)
7. The Division Algorithm for PolynomialsThe result is P(x) = D(x)Q(x) + R(x)If there is no remainder, then D(x) and Q(x) are factors of P(x)To check your answers, multiply D(x) and Q(x) then add R(x)
8. Example: Divide using long division. Checkyour answers.2 x +1 6 x + 7 x + 2 2
9. Example: Divide using long division. Checkyour answers.( 4x 2 + 23 x − 16 ) ÷ ( x + 5)
10. Example: Divide using long division. Check your answers.( 3x − 29 x + 56 ) ÷ ( x − 7 ) 2
11. Example: Divide using long division. Checkyour answers. (x 5 + 1) ÷ ( x + 1)
12. Checking FactorsTo check whether a polynomial is a factor of another polynomial, divide. If the remainder is zero, then the polynomial is a factor.
13. Example: Checking FactorsIs x 2 + 1 a factor of 3 x 4 − 4 x 3 + 12 x 2 + 5 ?
14. Example: Checking FactorsIs x 4 − 1 a factor of x 5 + 5 x 4 − x − 5 ?
15. Checking FactorsIf you need to check linear factors, we can use the factor theorem. Set the factor equal to zero and solve Plug the value into the other polynomial and simplify  If you get zero, then the factor you are checking is a factor of the polynomial
16. Example: Checking FactorsIs x − 2 a factor of P ( x ) = x 5 − 32 ? If it is, write P(x) as a product of two factors.