Adding Polynomials
Brenda Meery Jen Kershaw
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AUTHORS Brenda Meery Jen Kershaw
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C HAPTER
Chapter 1. Adding Polynomials
1
Adding Polynomials
In this concept, you will learn to add polynomials vertically and horizontally. Let’s Think About It
As the students rounded the corner on Fifth Street, they spotted a peculiar looking building. It was in the shape of a pyramid. Mrs. Meery, the math teacher, asked her students the following question. “A pyramid-shaped building has rectangular floors that get increasingly smaller as you go higher up in the building. If the 87th floor has a length of 6x + 16 and a width of 28, and each floor’s length and width decrease by 4 as you ascend, find the total area of the 87th , 88th , and 89th floor.” In this concept, you will learn to add polynomials. Guidance
A polynomial is an algebraic expression that shows the sum of monomials. In this concept you are going to add polynomials, but first let’s review how to add whole numbers with many digits. Add the numbers 5026 and 3210. You might choose to add it like this.
5026 +3210 8236
If you think about it, you might notice that the same addition could be thought of in this way. 1
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thousands 5026
→
+ 3210
→
8236
←
hundreds
tens
5000
ones
20
+3000 8000
200
10
+ 200
+ 30
6 +6
Each of the similar places has been lined up vertically (one on top of the other) so that 3000 is beneath 5000 in the thousands place and 10 is beneath 20 in the tens place. Also, 200 is by itself because the first number had no digits in the hundreds place. Likewise, 6 is by itself because the second number had no digits in the ones place. Although this is not a practical way of writing a simple addition problem, it does demonstrate the technique you can use to add polynomials. Polynomials can be added in the same manner as we added 5026 and 3210. You also need to know how to identify like terms. Like terms have exactly the same variable(s) to exactly the same power(s). When terms are alike, you can combine them by adding their coefficients. For example: 5x3 + 9x3 = 14x3 Let’s look at an example. Add the polynomials (7x2 + 9x − 5) and (6x2 + 3x + 10). First, line up the like terms so that you can add them vertically.
7x2 + 9x − 5
→
2
7x2
+
9x
+
−5
2
+
3x
+
10
+ 6x + 3x + 10 → + 6x
13x2 + 12x + 5 ← 13x2 +
12x +
5
Each of the like terms was aligned vertically, one on top of the other. Notice that the negative sign on -5 was kept with the number 5. Be careful when you add the integers. A second method for adding polynomials is horizontally—in a single line. Just as you might add 6 + 19 = 25 without placing them one on top of the other, polynomials can also be added horizontally. Let’s look at an example. Add the polynomials (7x2 + 3x − 11) and (3x2 − 9x + 5). First, rewrite the polynomials without parentheses. The polynomial can be rewritten without parentheses because the parentheses serve only to show the separation of the polynomials.
(7x2 + 3x − 11) + (3x2 − 9x + 5) = 7x2 + 3x − 11 + 3x2 − 9x + 5 Next, combine like terms.
(7x2 + 3x − 11) + (3x2 − 9x + 5) = 7x2 + 3x − 11 + 3x2 −9x + 5 = 10x2 −6x − 6 The answer is 10x2 − 6x − 6. 2
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Chapter 1. Adding Polynomials
Guided Practice
Add the polynomials (−2x3 + 9x2 − 3) and (8x5 + 5x − 14). First, rewrite the polynomials without parentheses.
(−2x3 + 9x2 − 3) + (8x5 + 5x − 14) = −2x3 + 9x2 − 3 + 8x5 + 5x − 14 Next, combine like terms. (−2x3 + 9x2 − 3) + (8x5 + 5x − 14) = −2x3 + 9x2 − 3 + 8x2 + 5x − 14 = −2x3 + 17x2 +5x − 17 The answer is −2x3 + 17x2 + 5x − 17. Examples Example 1
Add the polynomials (4x2 + 7x − 2) and (3x2 + 2x − 1). First, rewrite the polynomials without parentheses.
(4x2 + 7x − 2) + (3x2 + 2x − 1) = 4x2 + 7x − 2 + 3x2 + 2x − 1 Next, combine like terms. (4x2 + 7x − 2) + (3x2 + 2x − 1) = 4x2 + 7x − 2 + 3x2 +2x − 1 = 7x2 +9x − 3 The answer is 7x2 + 9x − 3. Example 2
Add the polynomials (−4x2 + 7x − 2) and (−7x2 + 3x − 17). First, rewrite the polynomials without parentheses.
(−4x2 + 7x − 2) + (−7x2 + 3x − 17) = −4x2 + 7x − 2 − 7x2 + 3x − 17 Next, combine like terms. (−4x2 + 7x − 2) + (−7x2 + 3x − 17) = −4x2 + 7x − 2 + −7x2 +3x − 17 = −11x2 +10x − 19 The answer is −11x2 + 10x − 19. 3
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Add the polynomials (4xy + 7x − 2) and (−19xy − 17x − 9). First, rewrite the polynomials without parentheses.
(4xy + 7x − 2) + (−19xy − 17x − 9) = 4xy + 7x − 2 − 19xy − 17x − 9 Next, combine like terms.
(4xy + 7x − 2) + (−19xy − 17x − 9) = 4xy + 7x − 2 + −19xy−17x − 9 = −15x2 −10x − 11 The answer is 15x2 − 10x − 11. Follow Up
Remember Mrs. Meery and the pyramids? First, write the expression for finding the area of the 87th , 88th , and 89th floor.
87th floor : Area = 28(6x + 16) 88th floor : Area = (28 − 4)(6x + 16 − 4) 89th floor : Area = (28 − 4 − 4)(6x + 16 − 4 − 4) Next, find the areas for the three floors. 4
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Chapter 1. Adding Polynomials
87th floor : Area = 28(6x + 16) : Area = 168x + 448 88th floor : Area = (28 − 4)(6x + 16 − 4) : Area = 24(6x + 12) : Area = 144x + 288 89th floor : Area = (28 − 4 − 4)(6x + 16 − 4 − 4) : Area = 20(6x + 8) : Area = 120x + 160 Then, find the total area for the three floors.
Total area = A87th floor + A88th floor + A89th floor = (168x + 448) + (144x + 288) + (120x + 160) = 432x + 896 The answer is 432x + 896. The total area for the 87th , 88th , and 89th floors 432x + 896 is units squared. Video Review
https://www.youtube.com/watch?v=KYZR7g7QcF4
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Explore More
Add the following polynomials vertically. Be sure to align like terms. 1. (4x2 + 7x − 2) + (3x − 17) 2. (−4x4 − x3 + 8) + (−2x3 + 5x + 6) 3. (10x3 − 4x2 − 2x + 5) + (−x2 + 9x − 5) 4. (6x2 + 5x + 9) + (4x2 + 3x + 6) 5. (9x2 − 3x + 4) + (6x2 − 9x + 2) 6. (3y2 + 4x − 9) + (−5y2 − 6x + 10) 5
www.ck12.org 7. (14x2 + 6x − 2) + (9x − 1) 8. (−2x2 + 7x − 2) + (−3x2 − 17) 9. (9x2 + 7x − 2y) + (3x2 − x + 9y) 10. (4xy + 7x − 21) + (−12xy + 4x − 8) 11. (11x2 + 9x − 2y) + (3x2 − 8x − 5y − 2) Add the following polynomials horizontally. 12. (−3x − 8) + (15x + 5) 13. (x4 + 7x3 − 2x + 7) + (−8x3 + 9x2 − 4) 14. (4x2 y − 3x2 y2 + 7xy) + (9x2 y2 − 5xy + 3x2 ) 15. (5xy − 3x + 19) + (4xy − 9x − 22)
References 1. Peggy2012CreativeLenz. https://www.flickr.com/photos/73230975@N03/6893326896/in/photolist-bv96VJe663bn-kEzmLN-6QoeNB-e6bFjU-5ykyQg-9gTmE-8QQNGQ-jEpmRC-5AYL8V-e3S38D-9uMUKq-9fWBAk -e9x9pU-9nBs2E-8pgk4a-5YmnFX-5MrhXc-4FhXup-cbcuFL-39Mjbe-48ovDd-939eyh-6T45kh-bMSu2-5Mvx jQ-58aJN3-mxLKP-avVZCv-mrRwQx-qUhbQf-j5b537-4BiMDn-cW9pQY-7u9izN-6byrb-96NbcG-5YUox3 -j572He-7KiA32-vbsWBW-7PzfAP-nxQ9P7-8QMHM4-bxJNhj-4yie8Q-9trkfH-mrRvzB-7tKs5-enfh4D .
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