Can you please give me the steps and answers? This is all in Solving partial fraction decomposition:

1. (13x-31) divided by (x^2-5x+6)

Decompose:

2. (13x^2-13x+22) divided by (x+2)(x^2+4)

Write the terms for:

3. (x+2) divided by x(x2+2x-35)

Solve by using inverse matrix

4. x+3y=-8
-14x-4y=-2

Help is greatly appreciated.

+1 vote

1). Take x2 - 5x + 6

Now solve the equation using the factor method.

x2 -3x - 2x + 6

x(x - 3) - 2(x - 3)

Take out common factors.

(x - 2)(x - 3)

(13x-31) / (x^2-5x+6) = (13x-31) / (x - 2)(x - 3)

Partial fractions:

(13x - 31) / (x - 2)(x - 3) = A/(x - 2) + B/(x - 3) ----------> (i)

Rewrite the expression with common denominator.

=[ A(x - 3) + B(x - 2)]/(x - 2)(x - 3)

= [Ax - 3A + Bx - 2B]/(x - 2)(x - 3)

(13x - 31) / (x - 2)(x - 3) = [(A + B)x- 3A - 2B]/(x - 2)(x - 3)

Compare the coefficients.

A + B = 13-----------> (1)

3A + 2B = 31--------->(2)

Multiply equation (1) by 2. and Subtract  equation (1), equation (2).

2A + 2B = 26

3A + 2B = 31

--------------------

-A = - 5

Multiply each side by negative one. Then A = 5

Substitute A = 5 in the equation (1).

5 + B = 13

Subtract 5 from each side.

B = 8.

Substitute A = 5 and B = 8 in the equation (i).

(13x - 31) / (x - 2)(x - 3) = 5/(x - 2) + 8/(x - 3).

+1 vote

2). Presumed (13x2 - 13x + 26) / (x + 2)(x2 + 4)

Partial fractions:

(13x2 - 13x + 26) / (x + 2)(x2 + 4) = A/(x + 2) + B/(x2 + 4) ----------> (i)

Rewrite the expression with common denominator.

=[ A(x2 + 4) + B(x + 2)]/(x + 2)(x2 + 4)

= [Ax2 + 4A + Bx + 2B]/(x + 2)(x2 + 4)

(13x2 - 13x + 26) / (x + 2)(x2 + 4) = [Ax2 +Bx + 4A + 2B]/(x + 2)(x2 + 4)

Compare the coefficients x2, x and constant.

x2 coefficients is A = 13, x coefficients is B = -13 and constant is 4A + 2B = 26

Substitute A = 13 and B = -13 in the equation (i)

(13x2 - 13x + 26) / (x + 2)(x2 + 4) = =13/(x + 2) - 13/(x2 + 4).

• 4).

The system of equations are :   x + 3y = - 8

- 14x - 4y = - 2

Write the equations in to matrix form .

Where, A  is coefficient matrix, X  is variable matrix and is constant matrix.

Variable matrix : .

If  , then it's inverse is .

.

.

The solution is x = 1 and y = - 3.

• 3).

Partial fraction decomposition for the rational function : .

First factor the denominator  .

.

Now, the rational function is .

Include one partial fraction for each factor.

.

Where, A, B, and C are to be determined.Multiplying this equation by least common denominator x(x + 7)(x - 5) yields the basic equation.

.

Compare coeffients and constants on each side.

.

.

Put,  in the above equations.

Solve eq (1) & (2) For, B and C.

.

Substitute the value in eq (1).

.

Therefore, .