Math 1110: Extra Credit Reading Assignment Attach this cover sheet to your work.
ID: 3116013 • Letter: M
Question
Math 1110: Extra Credit Reading Assignment Attach this cover sheet to your work. Be sure to answer all the questions completely on a separate page. I highly recommend reading all of the questions before answering any of them since often questions relate to each other. A monomial expressions is of the form ax", where a is any real number and n is a whole number (0, 1, 2, 3, etc). A polynomial expression is a sum of monomial expressions The degree of a given polynomial is the largest power of x and the leading coefficient of a polynomial is the constant associated with the term containing the largest power of x 1. a. Show that the following are polynomials and then state the degree and leading coefficient of each. ii. xV5-4 ii. (2x-5) Show that every linear function must be a polynomial and must either be degree 1 or 0. iv. x2 + 3x4 (2x)7 b. A 2. Suppose A and B are numbers in which = 0. a. b. What can you conclude about the value of A? Provide reasoning. What can you conclude about the value of B? Provide reasoning. A rational function is a function that can be put in the formf b(x) are polynomials and b(x) is not the zero polynomial. a(x) b(x) where af) and 3. Explain why bx) cannot be the zero polynomial. In other words, explain why b(x) cannot be zero for every input (x-value). 4 Show the following are rational functions by writing each as the ratio of two polynomials 2x2+1 x-1 2x+1 a. 4+ c. x2+5x+6Explanation / Answer
1 a. (i) 17 is a constant polynomial of degree 0.The leading coefficient is 17.
(ii) x 5 -43 is a polynomial of degree 1. The leading coefficient is 5.
(iii) On expansion, (x-2)(x-5) = x2-7x+10. It is a polynomial of degree 2. The leading coefficient is 1.
(iv) on rearranging, x2+3x4 –x6/3 –(2x)7 = -128x7- x6/3+3x4+ x2. It is a polynomial of degree 7. The
leading coefficient is -128.
b. As per its definition, linear functions are those whose graph is a straight line. A linear function has the form y = f(x) = mx+c. As may be observed, a linear function is a polynomial of degree 1 (when m 0) or of degree 0 (when m = 0).
2. a. If A and B are numbers such that A/B = 0, then on multiplying both the sides by B, we have (A/B)*B = B*0 or, A = 0.
b. B can be any number other than 0 as division by 0 is not defined.
3. b(x) cannot be the zero polynomial as division by 0 is not defined. In such a case a(x)/b(x) would be indeterminate.
4. a. We have 4 +(2x+1)/(x2+5x+6) = [4*(x2+5x+6)+(2x+1)]/ (x2+5x+6) = (4x2+22x+25)/ (x2+5x+6).
b. (1/x2+2/x -5) = (1-2x-5x2)/x2 = -(5x2+2x-1) and (x2/3- x4/2 +1) = (-x2+x2/3+1) = -(x2-x2/3-1) so that the given expression is -(5x2+2x-1)/-x2(x2-x2/3-1) = (5x2+2x-1)/(x4-x8/3-x2)
c. [(2x2+1)/(x-3)]-(x-1)/(x-4) = [(x-4)(2x2+1)-(x-1)(x-3)]/(x-3)(x-4) =[ (2x3-8x2+x-4) –(x2-4x+3)]/(x-3)(x-4) = (2x3-9x2+6x-7)/(x-3)(x-4) and [(x+5)/(x-3)]-[(x2+2)/(x-4)] = [(x+5)(x-4)- (x2+2)(x-3)]/(x-3)(x-4)= [(x2+x-20)-(x3-3x2+2x-6)]/(x-3)(x-4)=(-x3+4x2-x-14)/(x-3)(x-4) so that the given expression is (2x3-9x2+6x-7)/(-x3+4x2-x-14) = -(2x3-9x2+6x-7)/(x3-4x2+x+14).