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.

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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 $(x - 1)^{2} = 0 or (x - \sqrt{3}) = 0 or (x + \sqrt{3}) = 0$ Solve each of the above equations to obtain the zeros of $p (x)$ . $x = 1 (with multiplicity 2), x = \sqrt{3}, and x = - \sqrt{3}$

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

Make a Sign Table of Polynomials Questions with Solutions

Question 1

solution

Question 2

solution

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Make a Sign Table of Polynomials
Questions with Solutions