Question

What two rational expressions sum to 2x+3/x^2-5x+4

Enter your answer by filling in the boxes. Enter your answer so that each rational expression is in simplified form.

Answer 1

Explanation:

Given the rational expression: , to express this in simplified form, we would need to apply the concept of partial fraction.

Step 1: factorise the denominator

Thus, we now have:

Step 2: Apply the concept of Partial Fraction

Let,

=

Multiply both sides by (x - 1)(x - 4)

=

Step 3:

Substituting x = 4 in

Substituting x = 1 in

Step 4: Plug in the values of A and B into the original equation in step 2

Explanation:

Answer 2

`(2x + 3)/((x- 1)(x - 4)) = (-5)/(3(x- 1)) + (11)/(3(x - 4))`

Explanation:

Given the rational expression:
`(2x + 3)/(x^2 - 5x + 4)`, to express this in simplified form, we would need to apply the concept of partial fraction.

Step 1: factorise the denominator

`x^2 - 5x + 4`

`x^2 - 4x - x + 4`

`(x^2 - 4x) - (x + 4)`

`x(x - 4) - 1(x - 4)`

`(x- 1)(x - 4)`

Thus, we now have:
`(2x + 3)/((x- 1)(x - 4))`

Step 2: Apply the concept of Partial Fraction

Let,

`(2x + 3)/((x- 1)(x - 4))` =
`(A)/(x- 1) + (B)/(x - 4)`

Multiply both sides by (x - 1)(x - 4)

`(2x + 3)/((x- 1)(x - 4)) * (x - 1)(x - 4)` =
`((A)/(x- 1) + (B)/(x - 4)) * (x - 1)(x - 4)`

`2x + 3 = A(x - 4) + B(x - 1)`

Step 3:

Substituting x = 4 in
`2x + 3 = A(x - 4) + B(x - 1)`

`2(4) + 3 = A(4 - 4) + B(4 - 1)`

`8 + 3 = A(0) + B(3)`

`11 = 3B`

`(11)/(3) = B`

`B = (11)/(3)`

Substituting x = 1 in
`2x + 3 = A(x - 4) + B(x - 1)`

`2(1) + 3 = A(1 - 4) + B(1 - 1)`

`2 + 3 = A(-3) + B(0)`

`5 = -3A`

`(5)/(-3) = (-3A)/(-3)`

`A = -(5)/(3)`

Step 4: Plug in the values of A and B into the original equation in step 2

`(2x + 3)/((x- 1)(x - 4)) = (A)/(x- 1) + (B)/(x - 4)`

`(2x + 3)/((x- 1)(x - 4)) = (-5)/(3(x- 1)) + (11)/(3(x - 4))`