A26 APPENDIX Review To divide two polynomials, we first write each polynomial in standard form. The process then follows a pattern similar to that of Example 5. The next example illustrates the procedure. NOTE Remember, a polynomial is in standard form when its terms are written in descending powers of x. j Dividing Two Polynomials Find the quotient and the remainder when x x x x 3 4 7 is divided by 1 3 2 2 + + + + Solution EXAMPLE 6 Each polynomial is in standard form. The dividend is x x x 3 4 7, 3 2 + + + and the divisor is x 1. 2 + Step 1 Divide the leading term of the dividend, x3 ,3 by the leading term of the divisor, x .2 Enter the result, x3 , over the term x3 ,3 as follows: ) x x x x x 1 3 4 7 3 2 3 2 + + + + Step 2 Multiply x3 by x 1, 2 + and enter the result below the dividend. ) x x x x x x x 1 3 4 7 3 3 3 2 3 2 3 + + + + + Step 3 Subtract and bring down the remaining terms. ) x x x x x x x x x 1 3 4 7 3 3 4 2 7 3 2 3 2 3 2 + + + + + − + Step 4 Repeat Steps 1–3 using x x 4 2 7 2 − + as the dividend. ) x x x x x x x x x x x 1 3 4 7 3 3 4 2 7 4 4 2 3 3 4 2 3 2 3 2 2 + + + + + − + + − + + Since x2 does not divide x2− evenly (that is, the result is not a monomial), the process ends. The quotient is x3 4, + and the remainder is x2 3. − + Check: Quotient Divisor Remainder ⋅ + x x x x x x x x x x 3 4 1 2 3 3 3 4 4 2 3 3 4 7 Dividend 2 3 2 3 2 ( ) ( ) ( ) = + + + − + = + + + − + = + + + = Then x x x x x x x 3 4 7 1 3 4 2 3 1 3 2 2 2 + + + + = + + − + + ↑ Align the x3 term under the x to make the next step easier. ← Subtract (change the signs and add). ← Bring down the x4 2 and the 7. Divide x4 2 by x2 to get 4. ← Multiply ( ) + x 1 2 by 4; subtract. ( ) ← ⋅ + = + x x x x 3 1 3 3 2 3 COMMENT When the degree of the divisor is greater than the degree of the dividend, the process ends. ■ The next example combines the steps involved in long division.
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