Math A-100 IUSB Sec 7.3 Detailed notes – Week 12 Adding & Subtracting Rational Polynomials Since Rational polynomials are simply fractions with polynomials in the top and/or bottom, we can do anything with them that we can do with fractions. Recall from Section 6.1, we can multiply and divide them by first factoring them and cancelling. In Sec 6.2, we look at adding and subtracting rational polynomials. As with regular fractions, there are two different scenarios to work with: fractions with the same denominator and those with different denominators. Same Denominator Just as with regular fractions, if both fractions have the same denominator, make one numerator then add or subtract and reduce the new fraction. Reminder: in order to reduce a rational polynomial, you need to factor it completely to find what parts cancel out. Example 1: Add and simplify: 2y2 + y+1 5y + 3 y+1 Same denominator  2y2 + y+1 Factor & reduce  2y2 + 5y + 3 factors to (2y + 3)(y + 1) y+1 y+1 Cancel  5y + 3 = y+1 2y2 + 5y + 3 y+1 (2y + 3)(y + 1) Answer: 2y + 3 y+1 Example 2: Subtract and simplify: x2 _ 9x - 14 x –4 x2 – 4 2 Same denominator  x2 _ 9x - 14 becomes x2 - ( 9x – 14) x2 – 4 x2 – 4 x2 – 4 Clean it up  x2 - ( 9x – 14) x2 – 4 Factor  (x – 7)(x – 2) (x – 2)(x – 2) Cancel  (x – 7)(x – 2) (x – 2)(x – 2) becomes  x2 - 9x + 14 x2 – 4 Answer: x - 7 x-2 NOTE: make sure you distribute the “-“ sign to all terms in the 2nd fraction. Not doing co correctly creates the incorrect polynomial in the numerator, then thus will not reduce correctly. x2 + 4x - 21 x2 + 3x + 2 x2 + 3x + 2 Example 1: Add and simplify Same denominator  Factor  x2 + 4x - 21 becomes x2 + 3x + 2 x2 + 3x + 2 x2 + 4x - 21 x2 + 3x + 2 Does not reduce  x2 + 4x - 21 x2 + 3x + 2 (x – 3)(x + 7) (x + 1)(x + 2) becomes Answer: (x – 3)(x + 7) (x + 1)(x + 2) NOTE: just like regular fractions, some rational polynomials do not reduce. Special Cases “a – b & b – a” Recall from Section 7.1, we had the situation x – 4 4–x We cannot cancel these because the top & bottom are not exactly alike. We saw you can switch the order of terms by factoring out -1, giving you -1(4 – x) 4–x which now can be cancelled, -1(4 – x), leaving you with -1. 4–x 2y2 + y-1 Example 1: Add and simplify: 5y + 3 1–y Since the bottoms are not the same, we cannot simply add the fractions. However, since we have the a – b vs b – a situation, we can switch the order by factoring out -1. We can apply the – 1 to the top of the fraction, giving us bottoms that are the same. 2y2 + y-1  2y2 + y-1 5y + 3 1–y 5y + 3 -1(y – 1) Apply the -1 to the terms in the top of the fraction, giving you: 2y2 + y-1 Now add - 5y + - 3 y–1 2y2 + y-1 - 5y + - 3  2y2 - 5y - 3  (2y + 1)(y – 3) y–1 y–1 y–1