4.4 Dividing Polynomials (Objectives 1-4) Objective 1: Divide a Polynomial by a Monomial Divide a Polynomial by a Monomial To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and simplify the quotients using the rules for exponents. A B A BC C C or A B A BC C C NOTE: The idea of separating a fraction is used often in calculus. CAUTION: Be careful!!! 4.4.1 Divide. Objective 2: Divide Polynomials Using Long Division We can perform the long division of two polynomials in much the same way as the long division of two integers. For example, suppose we wish to divide the polynomial 22 7 4xx by the polynomial8x. The polynomial 22 7 4xx is the dividend and 8x is the divisor. We set up this problem using the familiar symbol used for long division with the dividend on the inside of the symbol and the divisor on the outside. 28 2 7 4x x x Step 1: Divide the first term of the dividend, 22x, by the first term of the divisor, x. Write the result, 2x, above the first term of the dividend. 228 2 7 4xx x x 222xxx or equivalently, 222x x x Step 2: Multiply 2x by 8x and line up like terms vertically. 2228 2 7 42 16xx x xxx 22 ( 8) 2 16x x x x Step 3: Subtract 22 16xx from 227xx. We perform the subtraction by changing the sign of each term of the polynomial being subtracted then combine like terms.2228 2 7 42 1623xx x xxxx Change the signs. 222 16 2 16x x x x Step 4: Bring down the constant 4. 2228 2 7 42 1623 4xx x xxxx Step 5: Divide 23x, by the first term of the divisor, x. Write the result, 23, above the second term of the dividend. 222 238 2 7 42 1623 4xx x xxxx 2323xx or equivalently, 23 23xx Step 6: Multiply 23 by 8x and line up like terms vertically. 222 238 2 7 42 1623 423 184xx x xxxxx 23( 8) 23 184xx Step 7: Subtract 23 184x from 23 4x. 222 238 2 7 42 1623 423 184188xx x xxxxx Change the signs. 23 184 23 184xx Note: If there are any missing terms in either polynomial, we must write both polynomials in standard form, inserting coefficients of 0 for any missing terms. 22 4 3170 3 0 4x x x xxx 4.4.11 Divide as indicated.2 4 3 0 5 6This is in .c x c 43These are the coefficients of 4 3 5 6.f x x x x 2 4 3 0 5 642 4 3 0 5 648Multiply 2 4=82 4 3 0 54 5682 4 3 0 5 6845102 4 3 0 5 68 1010452 4 3 0 5 68 10 204 255 102 4 3 0 5 68 20104 5 102 4 3 0 5 68 10 20455010 252 4 3 0 5 68 10 20 5045 4410 25Objective 2: Divide Polynomials Using Synthetic Division If a polynomial ()fxis divided by ()d x x c, then we can use a “short cut method” called synthetic division to find the quotient, ()qx, and the remainder, r. (Use this notation rather than what is written in the book.) You can see how much more compact the synthetic division process is compared to long division. Step 1: The constant coefficient, c, of the divisor ()d x x cis written to the far left while all coefficients of the polynomial ()fx are written inside the symbol . Once again, be sure to include the 0 for 20x. Step 2: Bring down the leading coefficient 4 Step 3: Multiply c times the leading coefficient that was just brought down. In this case we multiply 24. The product (8 in this case) is written in the next column in the second row. Step 4: Add down the column and write the sum (5 in this case) in the bottom row. We now repeat this process multiplying 2c times the value in the last row, always adding down the columns. Mulitiply 2 15 0 Add 0+10=10 Mulitiply 2 210 0 Add 5+20=25 Mulitiply 2 525 0 Add -6+50=442 4 3 0 5 68 10 20 504 5 10 25 44The last row now represents the quotient and the remainder. The last entry of the bottom row (44 in this case) is the remainder. The other numbers of the last row represent the coefficients of the quotient. remainder 44r coefficients of 32( ) 4 5 10 25q x x x x NOTE: synthetic division can only be used when the divisor has the form xc. 4.4.21 Divide using synthetic division. Objective 4: Divide Polynomial Functions. with Domain all values of x for which 4.4.25 Find and state any values that cannot be included in the domain of
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