How to Make a Sign Table of Polynomials
Questions with Detailed Solutions
How to make a sign table for polynomials? Grade 12 maths questions are presented along with detailed solutions and graphical interpretations.

Question 1Polynomial 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. solutionWe first find the zeros of the polynomial function 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. . 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. .
Question 2f(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.solutionWe 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. . 
More References and links
Introduction to PolynomialsFind Zeros of Polynomial Functions
Polynomial Questions and Problems with Solutions
High School Math (Grades 10, 11 and 12)  Free Questions and Problems With Answers
Middle School Math (Grades 6, 7, 8, 9)  Free Questions and Problems With Answers
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