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Chapter 3: Problem 102

Write a polynomial inequality whose solution set is \([-3,5]\)

Short Answer

Expert verified
The polynomial inequality whose solution set is \([-3,5]\) is \((x+3)(x-5) \leq 0\).

Step by step solution

01

Express the range as an inequality

Start the process by expressing the range (-3 to 5) as an inequality. It would be written as \(-3 \leq x \leq 5\) as it includes both -3 and 5.
02

Formulate the polynomial inequality

To convert this inequality into a polynomial inequality, you could use the expression \((x+3)(x-5)\). When expanded this gives \(x^2 -2x -15\). Moreover, because it should hold for the values between and including -3 and 5, the inequality should be \((x+3)(x-5) \leq 0\). That ensures that the values within that range will give a product that is less than or equal to zero.
03

Verifying the solution

The validity of the solution could be verified by substituting a number between -3 and 5 into the inequality. If the inequality holds true, then the solution is correct. For example, if x is 0, then the left-hand side of the inequality becomes \((-3)(-5)=15\) which is indeed bigger than 0, so the inequality holds true.

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Most popular questions from this chapter

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has no vertical, horizontal, or slant asymptotes, and no x -intercepts.

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