Based on material learned earlier in the course. The purpose of these problems is to keep the ma

Based on material learned earlier in the course. The purpose...

Key Concepts

Critical Points and Interval Testing

Critical points, which are the zeros of the polynomial, serve as the boundary points between intervals where the polynomial's sign may change. Interval testing involves choosing representative values from each interval to determine whether the polynomial expression is positive or negative in that region. This process is essential in determining the solution set for an inequality.

Factoring Polynomials

Factoring polynomials is the process of expressing a polynomial as a product of simpler polynomials or factors. This method is crucial when solving polynomial equations and inequalities because it allows for the identification of the roots or zeros of the function, which are essential for dividing the number line into intervals for sign analysis.

Polynomial Inequalities

Polynomial inequalities involve finding the set of values that satisfy an inequality where a polynomial expression is compared to zero or another polynomial. In solving these inequalities, one typically identifies the points where the polynomial equals zero, as these points separate the number line into intervals where the expression may be positive or negative.

Rearranging Inequalities

Rearranging inequalities is a fundamental technique in algebra where all terms are moved to one side of the inequality so that the expression is compared to zero. This step not only simplifies the problem but also sets the stage for further analysis, such as factoring and determining the sign of the expression over various intervals.

Transcript

00:01 So we have this inequality here, and the first thing i would do to solve it is get all of our terms on one side.

00:06 So i'm going to minus this 4x and this 20 over.

00:08 We have x of the third plus 5x squared minus 4x, minus 20 is now less than or equal to 0.

00:16 And now we can factor out an x squared in these first two terms and a negative 4 in the other two terms.

00:24 So we have negative 4 times x plus 5 as well.

00:28 And now we can factor out this x plus 5 term.

00:31 X plus 5 multiplied by x squared minus 4 so that's going to 0 and x squared minus 4 is a difference of squares so we can factor that even further we'd have x plus 5 times x plus 2 times x minus 2 and the way you factor difference of squares you just have the square root of the first square plus the square root of the second square multiplied by the square of the first square minus the square of the second square.

01:01 So this is less than or equal to zero.

01:03 So that means that if we let f of x equal this side of our inequality, then f of x is going to be equal to 0 at x is equal to negative 5, negative 2, and 2.

01:16 So we're going to want to look at values of x, where x is less than negative 5, where x is between negative 5 and negative 2, where x is between negative 2 and 2, and then when x is greater than 2.

01:27 So what x is less than negative 5, let's just pick negative 6.

01:31 We have negative 6 minus 5, or plus 5, sorry, which is a negative number.

01:35 Negative 6 plus 2 is also negative, and negative 6 minus 2 is also negative.

01:40 And a negative number multiplied by a negative number, and then multiplied by a negative number again is negative.

01:45 So that means if we let f of x equal this side of our inequality, f of x can be less than 0 when x is less than negative 5.

01:55 So now let's look at x between negative 5 and negative 2.

01:59 So we could pick the value of negative 4.

02:04 So if we have negative 4, negative 4 plus 5 is positive...

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