Make a Sign Table of Polynomials
Questions with Solutions

Master Grade 12 math with this step-by-step guide on creating sign tables for polynomial functions. This resource includes challenging practice questions, detailed solutions, and clear graphical interpretations to help you fully understand polynomial behavior.

Free Practice for SAT, ACT
and Compass Maths tests

Question 1

Polynomial p is given by \[ p(x) = (x - 1)^2(x - \sqrt 3) (x + \sqrt 3) \] Make a sign table of p and sketch a possible graph for \( p \).

solution

We first find the zeros of the polynomial function \( p(x) \). \[ p(x) = (x - 1)^2 (x - \sqrt{3}) (x + \sqrt{3}) = 0 \]

For \( p(x) = 0 \), we need to have \[ (x - 1)^2 = 0 \quad \text{or} \quad (x - \sqrt{3}) = 0 \quad \text{or} \quad (x + \sqrt{3}) = 0 \] Solve each of the above equations to obtain the zeros of \( p(x) \). \[ x = 1 \;\; \text{(with multiplicity 2)}, \quad x = \sqrt{3}, \quad \text{and} \quad x = -\sqrt{3} \]

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 \( x = 1 \)

\( x - \sqrt{3} > 0 \) for \( x > \sqrt{3} \)

\( x + \sqrt{3} > 0 \) for \( x > -\sqrt{3} \)


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.

Question 2

\( f(x) \) is a polynomial of degree six with a negative leading coefficient \( k \). \( f \) 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 References and links