Polynomial Review Problems 1. Find polynomial function formulas that could fit each of these graphs. Remember that you will need to determine the value of the leading coefficient. The point (0,-3) is on the graph The point (1, -10) is on the graph 2. The grid below shows a portion of the graph of a 6th degree polynomial, f(x). There may be additional x-intercepts that can t be seen in this graphing window. a. From the graph, list all of the factors of f(x) that can be determined. (0,0) b. What is the remainder when f(x) is divided by (x 1)? c. How many x-intercepts might f(x) have? Consider what the graph of f(x) might look like in a larger graphing window. In each of the boxes below, sketch a possible graph having the specified number of x-intercepts, or write impossible if that number of x-intercepts is not possible. 2 x-intercepts 3 x-intercepts 4 x-intercepts x-intercepts 6 x-intercepts d. Now, for the cases that you said were impossible, justify your answer.
3. a Use Lagrange interpolation to find a possible formula for the table below. You do not need to put your answer in normal form. x f (x) 1-9 2-4 4 6 b. Find a different possible formula for the table in part a. You do not need to put your answer in normal form. f ( x) 4. Suppose that a polynomial division problem has a quotient q(x) with remainder r(x). g( x) a. How is the degree of r(x) related to the degree of one of the other polynomials? b. Find an expression for f( 3) in terms of the other polynomials in this problem. c. Now suppose that g(x) = x + 3 and f(x) = 4x 3 7x 2 + 10. Divide to find q(x) and r(x).. a. Suppose f (x) = x 3 + ax 2 x 10. If the remainder when f (x) is divided by x 3 is 20, find the value of a. b. Suppose g(x) = 2x 3 x 2 + bx +14. If the remainder when g(x) is divided by x +1 is 18, find the value of b. 6. Find the value of k, such that x + 3 is a factor of: a. x + kx 2x 6 b. kx 3 2x 2 + x 6 7. Factor each polynomial over Z. a. 4 2 2x + 9x 18 b. 4x 2 2x 20 c. x 4 + x 2 + 4 d. 7x 8 + 21x +14x 2 e. x 6 1 f. 3x 3 + 6x 2 4x 8 8. Solve each of the following equations. Find all real solutions. a. x 4 + 63 =16x 2 b. x 3 + 2x 2 = 7x c. x 3 + x 2 = 7x + 3 d. x 3 + x 2 8x 6 = 0 (One solution is x = 3) e. x 3 7x = 0 f. (x 3)(2x 2 + 3x 6) = 0
9. Sketch graphs labeling zeros and the y-intercept. a. y = (x + 3)(x +1)(x 1) 2 b. y = 1 18 (x + 2)2 (x 1)(x 3) 2 10. Perform the following division: 4 x + 3x x + 2x + 11 a. 2 x 3x 1 b. 3 x + 8 x + 2 11. Without doing long division show that (x+1) is a factor of x 21 + x 20 + x 17 + x 16 + x 13 + x 12 + x 9 + x 8 + x + x 4 + x +1 12. Suppose f ( x) = x x 10x 8 and f (4) = 0. Sketch the graph of f(x).
Review problem answers 1. a. 1 ) 3 ( x + 1) ( x 3 b. x 3 (x 3) 2. a. x 2, (x 3) b. f(1) = 2 c. Graphs with 3, 4, and x-intercepts are all possible. (To get 4 x-intercepts you ll need to include another just touching x-intercept.) d. Having just 2 x-intercepts is impossible because 6th degree end behavior must be up on both sides or down on both sides, so the graph must cross the x-axis a third time. Having 6 x-intercepts is impossible because they would produce 6 linear factors, but at x = 0 there is even multiplicity so there would have to be a 7th factor (another x), and it s impossible for a 6th degree polynomial to have 7 linear factors. 3. a. f (x) = 3(x 2)(x 4) + 2(x 1)(x 4) + (x 1)(x 2) b. Add +(x 1)(x 2)(x 4) to the answer in part a. 4. a. deg r(x) < deg g(x). (If it weren t, you could continue dividing.) b. f( 3) = g( 3) q( 3) + r( 3) from the Euclidean Property. c. q(x) = 4x 2 19x + 7, r(x) = 161.. a. a = 2 b. b = 7 6. a. k = 3 b. k = 1 2 2 7. a. ( 2 3)( x + 6) x b. 2(2x )(x + 2) c. (x 2 +1)(x 2 + 4) d. 7x 2 (x 3 + 2)(x 3 + 1) e. (x 3 1)(x 3 + 1) f. (3x 2 4)(x + 2) 8. a. 3, -3, ± 7 b. 0, 1 ± 2 2 c. -, ± 7 d. x = 3, x =1± 3 e. 0, ±!! 9. a. b. 10 f. 3, 3 ± 7 4 10 10
10. a. x 2 + 6x +14 + 0x + 2 x 2 3x 1 2 b. x 2x + 4 11. Plug in -1 and the result is 0, so by the Factor Theorem (x+1) is a factor. 12.