Solving Polynomial Inequalities

In summary, a polynomial inequality is an inequality that contains a polynomial on one side and a constant on the other side, with a variable in between. To solve a polynomial inequality, you must simplify the inequality, set it equal to zero, factor the polynomial, use the zero product property, plot critical values on a number line, and test values in each interval to determine the solution set. The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. To check the solution for accuracy, plug in a value from each interval into the original inequality. While there are some shortcuts for certain types of polynomial inequalities, it is important to follow the standard process
  • #1
eleventhxhour
74
0
Solve the following inequality:

6e) $(x - 3)(x + 1) + (x - 3)(x + 2) \ge 0$

So, I created an interval table with the zeros x-3, x+1, x-3 and x+2 but I keep getting the wrong answer. Could someone help? (this is grade 12 math - so please don't be too complicated).

Thanks.
 
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  • #2
I would start with factoring the LHS:
$$(x-3)[(x+1)+(x+2)] \geq 0$$
$$\Leftrightarrow (x-3)(2x+3) \geq 0$$

Now create an interval table and look where the function $f(x)=(x-3)(2x+3)$ which has two zeros at $x=3$ and $x=\frac{-3}{2}$ is positive. That will give you the answer.
 

FAQ: Solving Polynomial Inequalities

What is a polynomial inequality?

A polynomial inequality is an inequality that contains a polynomial on one side and a constant on the other side, with a variable in between. It is written in the form ax^n + bx^(n-1) + ... + cx + d > 0 (or <, ≥, or ≤).

What is the process for solving polynomial inequalities?

To solve a polynomial inequality, follow these steps:

  1. Simplify the inequality by combining like terms.
  2. Set the inequality equal to zero by subtracting the constant from both sides.
  3. Factor the polynomial on the left side of the inequality.
  4. Use the zero product property to find the critical values.
  5. Plot these critical values on a number line and determine the intervals where the function is positive or negative.
  6. Test a value in each interval to determine the solution set.

What is the zero product property?

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is used to find the critical values of a polynomial inequality.

How do I know if my solution to a polynomial inequality is correct?

To check if your solution to a polynomial inequality is correct, you can plug in a value from each interval into the original inequality and see if it produces a true statement. If all values produce true statements, then your solution is correct.

Are there any shortcuts or tricks for solving polynomial inequalities?

There are some shortcuts for solving certain types of polynomial inequalities, such as using the quadratic formula for quadratic inequalities. However, it is important to follow the standard process for solving polynomial inequalities to ensure accuracy and avoid mistakes.

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