57 R.5 Polynomials When a polynomial has a missing term, we allow for that term by inserting a term with a 0 coefficient for it. For example, 3x2 - 7 is equivalent to 3x2 + 0x - 7, and 2x3 + x + 10 is equivalent to 2x3 + 0x2 + x + 10. EXAMPLE 10 Dividing Polynomials with MissingTerms Divide 3x3 - 2x2 - 150 by x2 - 4. SOLUTION Both polynomials have missing first-degree terms. Insert each missing term with a 0 coefficient. 3x - 2 x2 + 0x - 4)3x3 - 2x2 + 0x - 150 Missing term 3x3 + 0x2 - 12x -2x2 + 12x - 150 -2x2 + 0x + 8 12x - 158 Remainder The division process ends when the remainder is 0 or the degree of the remainder is less than that of the divisor. Because 12x - 158 has lesser degree than the divisor x2 - 4, it is the remainder. Thus, the entire quotient is written as follows. 3x3 - 2x2 - 150 x2 - 4 = 3x - 2 + 12x - 158 x2 - 4 S Now Try Exercise 103. R.5 Exercises CONCEPT PREVIEW Fill in the blank to correctly complete each sentence. 1. The polynomial 2x5 - x + 4 is a trinomial of degree . 2. A polynomial containing exactly one term is a(n) . 3. A polynomial containing exactly two terms is a(n) . 4. In the term -6x2y, -6 is the . 5. A convenient way to find the product of two binomials is to use the method. CONCEPT PREVIEW Perform the indicated operations. 6. 12x2 - x2 + 1x2 + 4x2 7. 4x1x3 - 7x2 8. -2x31x4 - 82 9. 1y + 422 10. 10m4 - 4m2 2m Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. See Example 1. 11. -5x11 12. -4y5 13. 6x + 3x4 14. -9y + 5y3 Missing term
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