Long DivisionWe 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
Example: Numerical Long DivisionDivide using long division. (Set up, Divide, Multiply, Subtract, Bring Down, Repeat) 672 ÷ 21
Polynomial Long DivisionDividing polynomials is useful when we are trying to factor polynomials, especially when we are unsure of factors.
The Division Algorithm for PolynomialsAn 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.
The Division Algorithm for PolynomialsYou 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)
The Division Algorithm for PolynomialsThe 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)
Example: Divide using long division. Checkyour answers.2 x +1 6 x + 7 x + 2 2
Example: Divide using long division. Checkyour answers.( 4x 2 + 23 x − 16 ) ÷ ( x + 5)
Example: Divide using long division. Check your answers.( 3x − 29 x + 56 ) ÷ ( x − 7 ) 2
Example: Divide using long division. Checkyour answers. (x 5 + 1) ÷ ( x + 1)
Checking FactorsTo check whether a polynomial is a factor of another polynomial, divide. If the remainder is zero, then the polynomial is a factor.
Example: Checking FactorsIs x 2 + 1 a factor of 3 x 4 − 4 x 3 + 12 x 2 + 5 ?
Example: Checking FactorsIs x 4 − 1 a factor of x 5 + 5 x 4 − x − 5 ?
Checking FactorsIf 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
Example: Checking FactorsIs x − 2 a factor of P ( x ) = x 5 − 32 ? If it is, write P(x) as a product of two factors.