Exponents and Polynomials - XYZ Custom Plus

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578Chapter 9 Exponents and PolynomialsMore about SequencesAs the next example indicates, when we substitute the counting numbers, inorder, into algebraic expressions, we form some of the sequences of numbersthat we studied previously. To review, recall that the sequence of counting numbers(also called the sequence of positive integers) isCounting numbers = 1, 2, 3, . . .8. Fill in the tablesa.n 1 2 3 43nExample 8Fill in the tables below to find the sequences formed bysubstituting the first four counting numbers into the expressions 2n and n 2 .a. b.n 1 2 3 4n 1 2 3 42nn 2b.n 1 2 3 4n 3Solution Proceeding as we did in the previous example, we substitute thenumbers 1, 2, 3, and 4 into the given expressions.a. When n = 1, 2n = 2 ⋅ 1 = 2.When n = 2, 2n = 2 ⋅ 2 = 4.When n = 3, 2n = 2 ⋅ 3 = 6.When n = 4, 2n = 2 ⋅ 4 = 8.As you can see, the expression 2n produces the sequence ofeven numbers when n is replaced by the counting numbers.Placing these results into our first table gives usn 1 2 3 42n 2 4 6 8b. The expression n 2 produces the sequence of squares when n isreplaced by 1, 2, 3, and 4. In table form we have.n 1 2 3 4n 2 1 4 9 16Answers8.a.n 1 2 3 43n 3 6 9 12Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Describe how to add polynomials.2. Describe how to subtract polynomials.3. Make up an addition problem involving two polynomials so that theanswer is the polynomial 8x 2 + 5x − 3.4. Find the value of the polynomial 8x 2 + 5x − 3 when x is 2.b.n 1 2 3 4n 3 1 8 27 64
9.2 Problem Set579Problem Set 9.2A Add the following polynomials. [Examples 1–3]1. (3x 2 + 2x + 5) + (3x 2 + 4x + 3)6x 2 + 6x + 82. (x 2 + 3x + 7) + (x 2 + 4x + 5)2x 2 + 7x + 123. (3a 2 − 4a + 2) + (2a 2 − 5a + 6)5a 2 − 9a + 84. (5a 2 − 2a + 1) + (3a 2 − 4a + 2)8a 2 − 6a + 35. (6x 3 − 5x 2 + 3x) + (9x 2 − 6x + 2)6x 3 + 4x 2 − 3x + 26. (5x 3 + 2x 2 + 4x) + (2x 2 + 5x + 1)5x 3 + 4x 2 + 9x + 17. (7t 2 − 4t − 5) + (6t 2 + 4t + 3)13t 2 − 28. (8t 2 − 3t − 2) + (7t 2 + 3t + 9)15t 2 + 79. (20x 3 − 12x 2 + x − 5) + (13x 3 + 2x 2 − 5x + 1)33x 3 − 10x 2 − 4x − 410. (30x 3 + 3x 2 − 10x + 3) + (12x 3 − 3x 2 + 10x − 3)42x 3B Subtract the following polynomials. [Examples 4–6]11. (4x 2 + 3x + 2) − (2x 2 + 5x + 1)2x 2 − 2x + 112. (8x 2 + 4x + 1) − (6x 2 + 2x + 3)2x 2 + 2x − 213. (9y 3 − 8y 2 − 4) − (5y 3 − 3y + 8)4y 3 − 8y 2 + 3y − 1214. (12y 2 − 2y + 9) − (10y 3 + 3y 2 − 9)−10y 3 + 9y 2 − 2y + 1815. (4x − 5) − (3x + 2) − (5x − 3)−4x − 416. (8x + 4) − (5x − 1) − (7x − 5)−4x + 1017. Subtract 10x 2 + 20x − 50 from 11x 2 − 10x + 14.x 2 − 30x + 6418. Subtract 2x 2 − 3x + 5 from 4x 2 − 6x + 12.2x 2 − 3x + 719. Subtract 3y 2 + 7y − 15 from 10y 2 + 10y + 10.7y 2 + 3y + 2520. Subtract 15y 2 − 7y + 3 from 15y 2 − 7y − 3.−6C Evaluate the following expressions for the given value. [Example 7]21. Find the value of x 2 − 10x + 25 when x is 2.922. Find the value of (x − 5) 2 when x is 2.923. Find the value of a 2 + 6a + 9 when a is −3.024. Find the value of (a + 3) 2 when a is −3.025. Find the value of 4y 2 + 12y + 9 when y = −1.126. Find the value of (2y + 3) 2 when y = −1.1
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