Sect. 1.3: Factoring


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1 Sect. 1.3: Factoring MAT 109, Fall 2015 Tuesday, 1 September 2015 Algebraic epression review Epanding algebraic epressions Distributive property a(b + c) = a b + a c (b + c) a = b a + c a Special epansion properties (A + B) (A  B) = A 2  B 2 (A + B) 2 = A A B + B 2 (A  B) 2 = A 22 A B + B 2 (A + B) 3 = A A 2 B + 3 A B 2 + B 3 (A  B) 3 = A 33 A 2 B + 3 A B 2  B 3 Use the above to epand the following algebraic epressions. 1. ( + 1) (  2) 2. (  4) ( + 4)
2 2 91_1.3_factoring.nb Consider the algebraic epression (  1) (  2) ( + 2). Epand this to answer the following questions. 1. This is a polynomial. What is its degree? 2. What is the coefficient of the 2 term? Epand the following to get two more special formulas. (A  B) A 2 + A B + B 2 (A + B) A 2  A B + B 2 Why Factor? Cubic splines In computer graphics and graphic design, curves are drawn using third degree polynomials, called cubic splines. The various polynomials must piece together so that there are no gaps. In other words each polynomial in the sequence of pieces must intersect with its neighbor polynomial. How would you determine for which values of the lines y = and y = intersect? What is the point of intersection of these two lines? How would you determine for which values of the cubic polynomials y = and y = intersect? Profit A company manufactures and sells iphone covers. Their manufacturing plant building cost $520,000. The materials for each cover cost $4.20. Also, personnel and energy costs vary depending on the quantities made. The manufacturing engineer approimates that to make covers, personnel and energy costs will be dollars. The company sells each cover for $ What is an algebraic epression that gives the profit for making and selling covers? How could one use this epression to determine the break even point, i.e. the number of covers to make and sell above which the company makes a profit, and below which it suffers a loss?
3 91_1.3_factoring.nb 3 Inverse operations What number added to 5 is 19? What is the opposite operation to addition? What number multiplied by 4 is 48? What is the opposite operation to multiplication? What is the number whose square is 81? What is the opposite operation to squaring? Factoring Factoring is the inverse operation to epanding a polynomial. Factoring is a useful way to find solutions to algebraic equations. Technique 1 Pull out what is common to all terms. (This again is the distributive property.) For eample, to factor the epanded epression 4 26, each term has factor = 2 (23) How does 3 h h 2 factor? How does 5 a 2 b 37 a 4 b 2 factor? Technique 2 We have some special epansion formula that we can use in reverse to factor particular epressions. (A + B) (A  B) = A 2  B 2 (A + B) 2 = A A B + B 2 (A  B) 2 = A 22 A B + B 2 (A + B) 3 = A A 2 B + 3 A B 2 + B 3 (A  B) 3 = A 33 A 2 B + 3 A B 2  B 3 (A  B) A 2 + A B + B 2 = A 3  B 3 (A + B) A 2  A B + B 2 = A 3 + B 3
4 4 91_1.3_factoring.nb How does factor? How does factor? How does y factor? Technique 3 Use trial and error focusing on the coefficients of the highest degree term and the constant term. Eample To factor recognize that the constant term 1 can be written as 1 (1). The coefficient of the highest degree term 2 2 is 2 and the simplest way to write 2 as a product is 1 2. Try epanding each of (21) ( + 1) (2 + 1) (  1) to see if one of them is If so, you have the factorization. What is the factorization of ? What is the factorization of ? What is the factorization of ? (Combine above techniques) Technique 4 Recognize the epression as a simpler epression after a replacement is made. Eample To factor (b + 2) (b + 2) + 5, not that this is the same as with = b + 2. Factor as ( + 5) ( + 1) using the third technique and put back b + 2 in place of to get (b + 2) (b + 2) + 5 = ((b + 2) + 7) ((b + 2) + 1) = (b + 9) (b + 3) What is the factorization of (31) 24 (31) + 3? What is the factorization of (5 y + 7) 2  (2 y  3) 2?
5 91_1.3_factoring.nb 5 Technique 5 Try factoring by breaking the epressions into groups. Eample To factor , recognize that the first two terms are similar to the last two terms. Factor out what is common to just the first to terms = 3 (  4) + ( + 4) Now factor out the common  4 factor = 3 (  4) + ( + 4) = ( + 4) Recognize the last cubic factor as one of the special forms, a sum of cubes = 3 (  4) + ( + 4) = ( + 4) ( + 4) ( + 1) Factor 4 y 32 y Factor 2 3/ / /2. (Hint: when you factor out common terms, think of the term with the smallest eponent, 1/2 as a common term.) 3. Mowing a field: A square field in a certain state park is mowed around the edges every week. The rest of the field is kept unmowed to serve as a habitat for birds and small animals. The field measures b feet by b feet, and the mowed strip is feet wide.
6 6 91_1.3_factoring.nb b b 3.1. Why is the area of the mowed portion b 2  (b  2 ) 2? 3.2. Factor this epression to come up with an equivalent epression for the mowed area. Homework imath problems on section 1.3b due by Saturday, September 5. Weekly assignment 2 due Thursday, September 3.
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