Lesson 6: Exponents and Polynomials Digging Deeper solutions

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1 Patterns in Products For this activity, we are going to explore patterns in products. Calculate the products below. Note: Feel free to use your calculator for this part of the activity Describe some of the patterns you notice in these products. Be as specific as possible and write down as many patterns as you can. In each case, the last two digits are 5. The difference between the first two products is 00, the second two is 400, the third two is 600, and so on. The hundreds and thousands digits are calculated by taking the tens digit of the original number and multiplying it by the next consecutive integer. For example, Multiply the tens digit (7) by the next integer (8) and you get 56, which are the thousands and hundreds digits of the product. If you look at the order that they are listed on the table, the first row differs by 3000, the second row by 4000, the third by 5000, and so on. 009 Duke University Talent Identification Program Page 1 of 5

2 Did you notice that if you ignore the tens and ones digits in the product (5 in each case), there is a pattern in the other digits? Can you figure out how to find the square of a random odd multiple of 5, without using a calculator? For example, there is a way to quickly figure out that the square of 45 is (When you know the trick, you should be able to do that one in your head in about 1 second). Can you figure out the pattern? In the space below, write the trick for squaring multiples of 5. For the example, 45 45, you can multiply 4 by the next integer (5) and you get 600. Then the product is Once you think you have the pattern, try it out for various squares. For example, try to square 145, 195, and 05 in your head. Now you can amaze your friends. Tell them, Give me any number that ends with a 5, and I can tell you its square. (Of course, you might want to suggest they keep the number to 3 digits or so!) This is a cool trick, and you can amaze your friends for hours using it! But mathematicians are not content with knowing that a trick is cool. We have to know why it works, and if it always works. In this activity, you ll use the formula for squaring a binomial to prove this trick always works. Recall the formula for squaring binomial is a b a ab b Start with a multiple of 5 (for example, 65) and write it in the form a+ b, where 5 a+ b and see if you can determine why your pattern b. Square works. Once it works for that number, try it with a new one until you have figured out the general pattern. In the space on the next page, explain what your pattern is and why it works. 009 Duke University Talent Identification Program Page of 5

3 Since the number is of the form ( a+ 5), we get: ( a ) a ( a) a + ( a) + a( a+ ) Since a is a multiple of 10, we can write a as 10x, where x is an integer. Then a x ( x + 1). So then the product above is: ( a+ 5) a( a+ 10) + 5 x ( x ) x( x ) 10 (10 + 1) When we add 100 x ( x + 1) ( 1) then x 7, x + 1 8, and ( 1) 56 x( x + 1) 56 and 5 is the number 565. and 5, we get the concatenation of x x + and 5. For example, if the number we are squaring is 75, x x +. The concatenation of There are other cool patterns in products, as well. For example, consider the products below: 009 Duke University Talent Identification Program Page 3 of 5

4 You ll notice that these form a similar pattern to that of the squares of multiples of 5. But these obviously are not squares. Why does this pattern hold for these numbers? Are there other products that follow this same pattern? In the space below, explain which pairs of numbers follow this pattern and why they do. If we write the first set of numbers as ( a 3)( a 7) get: ( a )( a ) 3 7 a 7a 3a a + 10a+ 1 a a As in the previous explanation, this can be written as: ( a+ )( a+ ) a( a+ ) + x( x ) , we FOIL this and The second set of numbers can be written as ( a 4)( a 6) becomes ( a 4)( a 6) a( a 10) 4 + +, and it As long as the ones digits add to equal 10 (and make the middle term 10a), this pattern holds. 009 Duke University Talent Identification Program Page 4 of 5

5 There is a different pattern for the products below: Lesson 6: Exponents and Polynomials See if you can find the pattern and use one of the multiplication formulas to justify the pattern. In the first set of numbers, all of the values are 4 less than a perfect square. For example, , and Since 4 is also a perfect square, we can factor the difference since it s a difference of squares. For the first one, we get: ( 0 )( 0 ) ( 18) + So each of the numbers can be written as the sum and difference of two numbers. The next one could be written as: Similarly, the second set of numbers can be written as ( a+ 5)( a 5) a+ 5 a 5 a 5. and in each case, we d get 009 Duke University Talent Identification Program Page 5 of 5