Polynomials - Nested Form

Category: Algebra

"Published in Newark, California, USA"

Rewrite the given polynomial into nested form 

If x = 1, find the value of the function.

Solution:

It is possible to eliminate the exponential powers of the given polynomial by rewriting it into nested form. Nested means grouped then factored. Let's consider the given equation above

Arrange the given polynomial according to their descending power if they are not yet arranged. If any of the exponential term is missing, you must include it by adding zero times the missing exponential term. Since the given polynomial is already arranged according to their descending power, then we can proceed with the grouping and factoring as follows

Therefore, in nested form is

If x = 1, then the value of the function is

Therefore, 

Check:

More Integration Procedures, 5

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate

Solution:

The first thing that we have to do is to find the differential of the given equation above. If

then

Hence, the above equation becomes

but

Substitute the value of sec² x to the above equation, we have

Therefore,

Partial Fractions

Category: Algebra

"Published in Newark, California, USA"

Resolve into partial fractions for

Solution:

In this lesson, we need to learn this one because you will use this method often when you will study or take-up Integral Calculus. If you will integrate the algebraic fractions, then you must rewrite it into partial fractions first. If you know how to combine a fraction either by addition, subtraction, multiplication, or division, then you must know how to split a fraction into partial fractions. Anyway, let's consider the given equation above

The first thing that we have to do is to factor the numerator and denominator if you can. You must simplify a fraction into  lowest term always. That's a rule in Mathematics. As you notice that one of the factor in the denominator which is x² + 3x + 3 cannot be factored, then we have to leave it as is. In this case, we have to split the above equation into partial fractions as follows

Multiply both sides of the equation by their Least Common Denominator (LCD) which is (2x + 3)(x² + 3x + 3) as follows

Expand the right side of the equation and group according to their variables 

In order to solve for the value of A, B, and C, we need to equate the both sides of the equation according to their variables.

For x²: 0 = A + 2B
           A = - 2B                         (equation 1)

For x: 12 = 3A + 3B + 2C
          12 = 3(- 2B) + 3B + 2C
          12 = - 6B + 3B + 2C
          12 = - 3B + 2C                (equation 2)

For x⁰: 21 = 3A + 3C
             7 = A + C
             7 = - 2B + C                 (equation 3)         

Use equation 2 and equation 3 to solve for the value of B as follows

      - 3B + 2C = 12     —————> - 3B + 2C = 12                  
    -2(- 2B + C = 7)                             4B - 2C = - 14
                                                    ————————
                                                                  B = - 2

Substitute the value of B to equation 3, we have

         - 2B + C = 7
     - 2(- 2) + C = 7
              4 + C = 7
                    C = 3

Substitute the value of B to equation 1, we have

                    A = - 2B
                    A = - 2(- 2)
                    A = 4

Therefore,