express x×x+x+4/(1-x)(x×x+1) into partial fraction
Assuming parentheses around the denominator have been omitted,
the expression is
x²+x+4/[(1-x)(x²+1)]
partial fractions for the third term is required, namely
4/[(1-x)(x²+1)]
=-4/[(x-1)(x²+1)]
=A/(x-1)+(Bx+C)/(x²+1)
By adding the right-hand side, we have
A(x²+1)+(Bx+C)(x-1)=-4
By matching coefficients of x²,x and constant term, we obtain
A+B=0
B+C=0
A-C=-4
Solving, A=-2, B=C=2
thus the partial fraction is
(2x+2)/(x²+1)-2/(x+1)
Be sure to add the partial fractions to the first two terms x²+x.
And assuming the usual sloppiness in the numerator, we have
(x²+x+4)/[(1-x)(x²+1)]
= (2x+1)/(x²+1) + 3/(1-x)
amazing how little difference it made, eh?
To express the given expression as partial fractions, we need to decompose it into simpler fractions.
Step 1: Factorize the denominator (1-x)(x^2+1):
The denominator, (1-x)(x^2+1), cannot be factored further as it represents the product of two different irreducible quadratic factors.
Step 2: Write the expression in partial fraction form:
Since the denominator cannot be factored further, we can start with the partial fraction form:
(x^2 + x + 4) / ((1 - x)(x^2 + 1)) = A / (1 - x) + (Bx + C) / (x^2 + 1)
Step 3: Finding the values of A, B, and C:
To find the values of A, B, and C, we can multiply the entire equation by the common denominator (1 - x)(x^2 + 1) and simplify it:
(x^2 + x + 4) = A(x^2 + 1) + (Bx + C)(1 - x)
Expanding:
x^2 + x + 4 = Ax^2 + A + Bx - Bx^2 + C - Cx
Grouping the terms with similar powers:
(x^2 + x + 4) = (A - B)x^2 + (B - C)x + (A + C)
By equating the coefficients of each term, we get the following system of equations:
For x^2: A - B = 1
For x: B - C = 1
For constant: A + C = 4
Solving these equations simultaneously gives us the values of A, B, and C.
Step 4: Solving the system of equations:
From the first equation (A - B = 1), we get A = B + 1.
Substituting this into the third equation (A + C = 4), we get (B + 1) + C = 4, which simplifies to B + C = 3.
From the second equation (B - C = 1), we can substitute B + C = 3 into it, which gives us 3 - C = 1, simplifying to C = 2.
Substituting C = 2 into B + C = 3, we find B + 2 = 3, which gives us B = 1.
Finally, using A = B + 1, we get A = 1 + 1, which is A = 2.
Step 5: Writing the expression in partial fraction form:
Now that we have the values of A, B, and C, we can write the expression in partial fraction form:
(x^2 + x + 4)/(1 - x)(x^2 + 1) = 2/(1 - x) + (x + 2)/(x^2 + 1)
Therefore, the given expression can be expressed as the sum of two partial fractions: 2/(1 - x) + (x + 2)/(x^2 + 1).