# How to Solve Polynomial Inequalities

In this post, let’s look at how to solve inequalities involving polynomial.

Let’s start off by looking at the steps needed to solve the polynomial inequalities:

Step 1: Put all the non- zero terms on one side of the inequality, and make the other side of the inequality 0

Step 2: Completely factorize the side with the polynomial

Step 3: Determine the values that will make the polynomial 0.

Step 4: Put the values in step 3 on a number line.

Step 5: Determine whether the polynomial is positive or negative at the intervals on the number line.

Step 6: Determine the solution.

### Example 1: Solve the inequality (x-3)(x+2)(x-1)>(x-3)(x+1)

Step 1: Put all the terms to one side, and make the other side 0.

(x-3)(x+2)(x-1)-(x-3)(x+1)>0

Step 2: Completely factorize the side with the polynomial

(x-3)(x-1)[(x+2)-(x+1)]>0

(x-3)(x²-3) > 0

(x-3)(x+√3)(x-√3)>0

Step 3: Determine the values that will make the polynomial 0.

For the polynomial (x-3)(x+√3)(x-√3), it is equal to 0 when x = 3, -√3 or √3

Step 4: Put the values in step 3 on a number line.

Step 5: Determine whether the polynomial is positive or negative at the intervals on the number line.

I’ve marked out the 4 intervals that we are interested in, and labelled them A, B, C and D.

Let’s look at interval A. Interval A consists of x values that are lesser than -√3.

Let’s put any value of x lesser than -√3 into (x-3)(x+√3)(x-√3). E.g., let’s substitute x = -4 into it. (-4-3)(-4+√3)(-4-√3) =-91 < 0

Hence, we’ll put a negative sign at A.

Next, let’s look at interval B. Interval B consists of x values between -√3 and √3.

Let’s put any value of x between -√3 and √3 into (x-3)(x+√3)(x-√3). E.g., let’s substitute x = 0 into it. (0-3)(0+√3)(0-√3) =9 > 0.

Hence, we’ll put a positive sign at B.

Next, let’s look at interval C. Interval C consists of x values between √3 and 3.

Let’s put any value of x between √3 and 3 into (x-3)(x+√3)(x-√3). E.g., let’s substitute x = 2 into it. (2-3)(2+√3)(2-√3) =-1 < 0.

Hence, we’ll put a negative sign at C.

Finally, let’s look at interval D. Interval D consists of x values more than 3.

Let’s put any value of x more than 3 into (x-3)(x+√3)(x-√3). E.g., let’s substitute x = 4 into it. (4-3)(4+√3)(4-√3) =13 > 0.

Hence, we’ll put a positive sign at D.

Step 6: Determine the solution.

Since we are solving for (x-3)(x+√3)(x-√3)>0, we are interested in the range of values of x that makes (x-3)(x+√3)(x-√3) positive. From the number line obtained from step 5, we know that this occurs when -√3<x<√3 or x > 3 (see the region highlighted in yellow below).

Hence, the solution to (x-3)(x+2)(x-1)>(x-3)(x+1) is -√3<x<√3 or x > 3.

## Learn H2 A Level Math Inequalities

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## All the notes for H2 A Level Math

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