Special Products - Factoring, 2

Category: Algebra

"Published in Newark, California, USA"

Find the factors for

Solution:

The given equation above is a quadratic equation where the variables or terms are polynomials like (2x - 1) and (x + 3). 

To test the above equation if it is factorable or not, let's consider the following procedure: if a = 8, b = -2, and c = -3, then using the discriminant formula, we have

Since the value of discriminant is a perfect square, then the given equation is factorable. Consider the given equation above

Factor the above equation in terms of (2x - 1) and (x + 3), we have

Special Products - Rational Functions

Category: Algebra

"Published in Newark, California, USA"

 Write  the product and simplify for

 

Solution:

The given equation above is another good example of special products where a binomial multiplied by another binomial where the terms are rational functions. In this case, we have to do the distribution property of multiplication over addition as follows

The LCD (Least Common Denominator) of the three fractions is (x + y)²(a + b)². Rewrite the three fractions in terms of their LCD and simplify, we have

Since all the terms of the numerator cannot be combined and simplified, then we can leave the final answer as is.

Inverse Trigonometric Functions

Category: Trigonometry, Algebra

"Published in Suisun City, California, USA"

Rewrite the expression as an algebraic expression in x for 

Solution:

Consider the given equation above, we have

Let

and

so that the above equation becomes

Next, we need to get the values of trigonometric functions as a function of x. Consider again the given inverse trigonometric functions in order to get the trigonometric functions as a function of x as follows:

Let 

Draw and label a right triangle in order to get the other trigonometric functions as a function of x

Photo by Math Principles in Everyday Life

Let

Draw and label a right triangle in order to get the other trigonometric functions as a function of x

Photo by Math Principles in Everyday Life

Finally, consider again the function

Substitute the values of trigonometric functions to the above equation, we have

Rationalize the denominator in order to eliminate the radical sign

Therefore,