The ability to concentrate depends on adequate sleep decent nutritionand the

# The ability to concentrate depends on adequate sleep

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The ability to concentrate depends on adequate sleep, decent nutrition,and the phys- ical well-being that comes with exercise.
Solution a) y 9 5 y 4 Use the de fi nition of division to check that y 4 y 5 y 9 . b) 4 Use the de fi nition of division to check that 3 b 7 12 b 2 . c) 6 x 3 (2 x 9 ) Use the de fi nition of division to check that 2 x 9 6 x 3 . d) x 6 y 0 x 6 Use the de fi nition of division to check that x 6 x 2 y 2 x 8 y 2 . We showed more steps in Example 2 than are necessary. For division problems like these you should try to write down only the quotient. Dividing a Polynomial by a Monomial We divided some simple polynomials by monomials in Chapter 1. For example, (6 x 8) 3 x 4. We use the distributive property to take one-half of 6 x and one-half of 8 to get 3 x 4. So both 6 x and 8 are divided by 2. To divide any polynomial by a mono- mial, we divide each term of the polynomial by the monomial. E X A M P L E 3 Dividing a polynomial by a monomial Find the quotient for ( 8 x 6 12 x 4 4 x 2 ) (4 x 2 ). Solution 2 x 4 3 x 2 1 The quotient is 2 x 4 3 x 2 1. We can check by multiplying. 4 x 2 ( 2 x 4 3 x 2 1) 8 x 6 12 x 4 4 x 2 . Because division by zero is unde fi ned, we will always assume that the divisor is nonzero in any quotient involving variables. For example, the division in Exam- ple 3 is valid only if 4 x 2 0, or x 0. 4 x 2 4 x 2 12 x 4 4 x 2 8 x 6 4 x 2 8 x 6 12 x 4 4 x 2 4 x 2 1 2 6 x 8 2 y 2 y 2 x 8 x 2 x 8 y 2 x 2 y 2 6 x 9 x 6 3 x 6 3 x 6 6 x 3 2 x 9 12 b 7 b 5 4 b 5 4 b 5 1 b 7 2 b 2 b 7 12 3 12 b 2 3 b 7 y 9 y 5 232 (4-26) Chapter 4 Polynomials and Exponents h e l p f u l h i n t s t u d y t i p Play offensive math,not defen- sive math.A student who says, “Give me a question and I’ll see if I can answer it,” is playing defensive math. The student is taking a passive approach to learning. A student who takes an active approach and knows the usual questions and answers for each topic is playing offensive math. Recall that the order of oper- ations gives multiplication and division an equal ranking and says to do them in order from left to right. So without parentheses, 6 x 3 2 x 9 actually means x 9 . 6 x 3 2
Dividing a Polynomial by a Binomial Division of whole numbers is often done with a procedure called long division. For example, 253 is divided by 7 as follows: 36 Quotient Divisor 7 253 Dividend 21 43 42 1 Remainder Note that 36 7 1 253. It is always true that (quotient)(divisor) (remainder) dividend. To divide a polynomial by a binomial, we perform the division like long divi- sion of whole numbers. For example, to divide x 2 3 x 10 by x 2, we get the fi rst term of the quotient by dividing the fi rst term of x 2 into the fi rst term of x 2 3 x 10. So divide x 2 by x to get x , then multiply and subtract as follows: 1 Divide: x x 2 x x 2 Multiply: x 2 x 2 3 x 10 x 2 2 x x ( x 2) x 2 2 x 3 Subtract: 5 x 3 x 2 x 5 x Now bring down 10 and continue the process. We get the second term of the quo- tient (below) by dividing the fi rst term of x 2 into the fi rst term of 5 x 10. So divide 5 x by x to get 5: 1 Divide: x 5 5 x x 5 2

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