Determine if the fraction is proper or improper.

Question

ScanSolve AI Tutor · Accepted Answer

To resolve the given expression into partial fractions, we first check if it is a proper or improper fraction.

Determine if the fraction is proper or improper.
The given fraction is $\frac{x^2 + 1}{2x^2 - 3x + 2}$ .
The degree of the numerator ( $x^2+1$ ) is 2.
The degree of the denominator ( $2x^2 - 3x + 2$ ) is 2.
Since the degree of the numerator is equal to the degree of the denominator, the fraction is improper. We must perform polynomial long division first.
Perform polynomial long division.
Divide $x^2 + 1$ by $2x^2 - 3x + 2$ .
$\frac{x^2 + 1}{2x^2 - 3x + 2}$
We can write this as:
$x^2 + 1 = A(2x^2 - 3x + 2) + (Bx + C)$
Comparing the coefficients of $x^2$ :
$1 = 2A \implies A = \frac{1}{2}$
Substitute $A = \frac{1}{2}$ back into the equation:
$x^2 + 1 = \frac{1}{2}(2x^2 - 3x + 2) + Bx + C$ $x^2 + 1 = x^2 - \frac{3}{2}x + 1 + Bx + C$
Subtract $x^2 - \frac{3}{2}x + 1$ from both sides:
$0 = -\frac{3}{2}x + Bx + C$ $0 = \left(B - \frac{3}{2}\right)x + C$
Comparing the coefficients of $x$ :
$B - \frac{3}{2} = 0 \implies B = \frac{3}{2}$
Comparing the constant terms:
$C = 0$
So, the division gives a quotient of $\frac{1}{2}$ and a remainder of $\frac{3}{2}x$ .
Therefore, we can write the expression as:
$\frac{x^2 + 1}{2x^2 - 3x + 2} = \frac{1}{2} + \frac{\frac{3}{2}x}{2x^2 - 3x + 2}$
To simplify the remainder term, multiply the numerator and denominator by 2:
$\frac{x^2 + 1}{2x^2 - 3x + 2} = \frac{1}{2} + \frac{3x}{2(2x^2 - 3x + 2)}$
Check if the denominator of the remainder term can be factored.
The denominator of the proper fraction part is $2x^2 - 3x + 2$ .
We calculate the discriminant $\Delta = b^2 - 4ac$ for this quadratic.
Here, $a=2$ , $b=-3$ , $c=2$ .
$\Delta = (-3)^2 - 4(2)(2) = 9 - 16 = -7$
Since the discriminant $\Delta < 0$ , the quadratic $2x^2 - 3x + 2$ has no real roots and is irreducible over the real numbers.
State the partial fraction decomposition.
Since the denominator $2x^2 - 3x + 2$ is an irreducible quadratic factor, the term $\frac{3x}{2(2x^2 - 3x + 2)}$ is already in its simplest partial fraction form.
Thus, the partial fraction decomposition is:
$\frac{1}{2} + \frac{3x}{2(2x^2 - 3x + 2)}$

Determine if the fraction is proper or improper.

Answer

Step-by-step solution

Determine if the fraction is proper or improper.

Perform polynomial long division.

Check if the denominator of the remainder term can be factored.

State the partial fraction decomposition.

Try one yourself

Still stuck on this one?

More Mathematics Questions