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# 5.4 long division

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### Transcript

• 1. Part 1: Long Division
• 2. Long Division&#xF097;We can divide polynomials using steps that are similar to the steps of numerical long division a Notation: a &#xF7; b = = b a b Vocabulary: dividend &#xF7; divisor = quotient
• 3. Example: Numerical Long DivisionDivide using long division. (Set up, Divide, Multiply, Subtract, Bring Down, Repeat) 672 &#xF7; 21
• 4. Polynomial Long Division&#xF097;Dividing polynomials is useful when we are trying to factor polynomials, especially when we are unsure of factors.
• 5. The Division Algorithm for Polynomials&#xF097;An algorithm is a specific set of instructions used to solve a problem.&#xF097;The Division Algorithm for Polynomials is a generalized version of the technique of long division in arithmetic. &#xF097;To divide polynomials, list polynomials in standard form with zero coefficients where appropriate.
• 6. The Division Algorithm for Polynomials&#xF097;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). &#xF097;Set up, Divide, Multiply, Subtract (change signs), Bring Down, Repeat Q( x) D( x) P( x) O R( x) &#xF097;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&#xF097;The result is P(x) = D(x)Q(x) + R(x)&#xF097;If there is no remainder, then D(x) and Q(x) are factors of P(x)&#xF097;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 &#x2212; 16 ) &#xF7; ( x + 5)
• 10. Example: Divide using long division. Check your answers.( 3x &#x2212; 29 x + 56 ) &#xF7; ( x &#x2212; 7 ) 2
• 11. Example: Divide using long division. Checkyour answers. (x 5 + 1) &#xF7; ( x + 1)
• 12. Checking Factors&#xF097;To check whether a polynomial is a factor of another polynomial, divide. &#xF097;If the remainder is zero, then the polynomial is a factor.
• 13. Example: Checking Factors&#xF097;Is x 2 + 1 a factor of 3 x 4 &#x2212; 4 x 3 + 12 x 2 + 5 ?
• 14. Example: Checking Factors&#xF097;Is x 4 &#x2212; 1 a factor of x 5 + 5 x 4 &#x2212; x &#x2212; 5 ?
• 15. Checking Factors&#xF097;If you need to check linear factors, we can use the factor theorem. &#xF097;Set the factor equal to zero and solve &#xF097;Plug the value into the other polynomial and simplify &#xF097; If you get zero, then the factor you are checking is a factor of the polynomial
• 16. Example: Checking Factors&#xF097;Is x &#x2212; 2 a factor of P ( x ) = x 5 &#x2212; 32 ? If it is, write P(x) as a product of two factors.
• 17. Homework&#xF097;P308 #9 &#x2013; 19 odd, 44 &#x2013; 51 odd