Determine if the fraction is proper or improper.

Welcome back Benzema — missed you this week. To resolve the given expression into partial fractions, we first check if it is a proper or improper fraction. Step 1: Determine if the fraction is proper or improper. The given fraction is (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. Step 2: Perform polynomial long division. Divide x^2 + 1 by 2x^2 - 3x + 2. (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 A = (1)/(2) Substitute A = (1)/(2) back into the equation: x^2 + 1 = (1)/(2)(2x^2 - 3x + 2) + Bx + C x^2 + 1 = x^2 - (3)/(2)x + 1 + Bx + C Subtract x^2 - (3)/(2)x + 1 from both sides: 0 = -(3)/(2)x + Bx + C 0 = (B - (3)/(2))x + C Comparing the coefficients of x: B - (3)/(2) = 0 B = (3)/(2) Comparing the constant terms: C = 0 So, the division gives a quotient of (1)/(2) and a remainder of (3)/(2)x. Therefore, we can write the expression as: (x^2 + 1)/(2x^2 - 3x + 2) = (1)/(2) + (3)/(2)x2x^2 - 3x + 2 To simplify the remainder term, multiply the numerator and denominator by 2: (x^2 + 1)/(2x^2 - 3x + 2) = (1)/(2) + (3x)/(2(2x^2 - 3x + 2)) Step 3: 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 = b^2 - 4ac for this quadratic. Here, a=2, b=-3, c=2. = (-3)^2 - 4(2)(2) = 9 - 16 = -7 Since the discriminant < 0, the quadratic 2x^2 - 3x + 2 has no real roots and is irreducible over the real numbers. Step 4: State the partial fraction decomposition. Since the denominator 2x^2 - 3x + 2 is an irreducible quadratic factor, the term (3x)/(2(2x^2 - 3x + 2)) is already in its simplest partial fraction form. Thus, the partial fraction decomposition is: (1)/(2) + (3x)/(2(2x^2 - 3x + 2)) Send me the next one 📸