How to Make a Sign Table of Polynomials
Questions with Detailed Solutions

How to make a sign table for polynomials. Grade 12 math questions are presented along with detailed solutions and graphical interpretations.



Free Practice for SAT, ACT
and Compass Math tests

  1. Polynomoal p is given by $$p(x) = (x - 1)^2(x - √3) (x + √3) $$
    Make a sign table of p and sketch a possible graph for p.
    Solution
    We first find the zeros of p.
    p(x) = (x - 1)2 (x - √3) (x + √3) = 0
    For p(x) = 0, we need to have
    (x - 1)2 = 0 , or (x - √3) = 0 , or (x + √3) = 0
    Solve each of the above equations to obtain the zeros of p(x).
    x = 1 (multiplicity 2) , x = √3 and x = - √3
    c) With the help of the factored form of p(x) and its zeros found above, we now make a table of signs using:
    (x - 1)2 is positive for all x except at x = 1
    x - √3 > 0 for x > √3
    x + √3 > 0 for x > - √3
    We put each factor in the table and use the rules of multiplication of signs to complete the sign for p as shown below.

    table of sign question 3 .


    We use the zeros of p(x) which graphically are shown as x intercepts, the table of signs and the y intercept (0 , -3) to complete the graph as shown below.

    polynomials question 1 .


  2. f(x) is a polynomial of degree six with a negative leading coefficient. f has a zero of multiplicity 1 at x = -1, a zero of multiplicity 3 at x = 1, and a zero of multiplicity 2 at x = 3. Make a sign table for the polynomial f.
    Solution
    We first write the factors of polynomial f with their multiplicity.
    zero of multiplicity 1 at x = -1 : factor: x + 1
    zero of multiplicity 3 at x = 1 : factor: (x - 1)3
    zero of multiplicity 2 at x = 3 : factor: (x - 3)2
    Let k (negative) be the leading coefficient of f. Using all the above factors, we write f(x) as
    f(x) = k (x + 1)(x - 1)3(x - 3)2
    We first study the sign of the different factors of f.
    x + 1 > 0 for x > - 1
    (x - 1)3 > 0 for x > 1
    (x - 3)2 > 0 for all x except x = 3
    Below is shown the table of signs of each factor and of the polynomial f(x) in the bottom row.

    table of sign question 2 .

More High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers
More Middle School Math (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
More Primary Math (Grades 4 and 5) with Free Questions and Problems With Answers
Author - e-mail
Home Page