If (2x^3 - 7x^2 - 2x + 20) / (x^2 - 5x + 6) = Ax + B + C/(x - 2) + D/(x - 3), then A + B + C + D =

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q19 – AP-EAMCET 2026

← Back

Home Maths Complex & QuadraticQuestion #58

\frac{2 x^{3} - 7 x^{2} - 2 x + 20}{x^{2} - 5 x + 6} = A x + B + \frac{C}{x - 2} + \frac{D}{x - 3}

, then

A + B + C + D =

Clue 1 of 3: Rational Functions

When the degree of the numerator is greater than or equal to the degree of the denominator in a rational function, perform polynomial long division first.

🧠 Full Solution Path

Step 1: Perform Polynomial Long Division✦ Active

The degree of the numerator ( $2 x^{3} - 7 x^{2} - 2 x + 20$ ) is 3, and the degree of the denominator ( $x^{2} - 5 x + 6$ ) is 2. Since the degree of the numerator is greater than or equal to the degree of the denominator, we first perform polynomial long division.

b e g i n a r r a y r 2 x + 3 x^{2} - 5 x + 6 o v e r l i n e) 2 x^{3} - 7 x^{2} - 2 x + 20 u n d e r l i n e - (2 x^{3} - 10 x^{2} + 12 x) 3 x^{2} - 14 x + 20 u n d e r l i n e - (3 x^{2} - 15 x + 18) x + 2 e n d a r r a y

This division gives us a quotient of $2 x + 3$ and a remainder of $x + 2$ . Thus, we can write the given expression as:

\frac{2 x^{3} - 7 x^{2} - 2 x + 20}{x^{2} - 5 x + 6} = 2 x + 3 + \frac{x + 2}{x^{2} - 5 x + 6}

Comparing this to the given form $A x + B + \frac{C}{x - 2} + \frac{D}{x - 3}$ , we identify:

A = 2

B = 3

💡 Teacher's Secret Hint

Remember to handle subtraction carefully during polynomial long division, especially when signs change.

Step 2: Factor Denominator and Decompose Remainder○ Expand

Next, we need to decompose the remainder term $\frac{x + 2}{x^{2} - 5 x + 6}$ using partial fractions. First, factor the denominator:

x^{2} - 5 x + 6 = (x - 2) (x - 3)

Now, set up the partial fraction decomposition for the remainder term:

\frac{x + 2}{(x - 2) (x - 3)} = \frac{C}{x - 2} + \frac{D}{x - 3}

Multiply both sides by $(x - 2) (x - 3)$ to clear the denominators:

x + 2 = C (x - 3) + D (x - 2)

To find $C$ , substitute $x = 2$ :

2 + 2 = C (2 - 3) + D (2 - 2)

4 = - C \Rightarrow C = - 4

To find $D$ , substitute $x = 3$ :

3 + 2 = C (3 - 3) + D (3 - 2)

5 = D \Rightarrow D = 5

So, we have $C = - 4$ and $D = 5$ .

💡 Teacher's Secret Hint

The 'cover-up method' (Heaviside's method) can be a quick way to find constants in simple linear partial fractions. For example, to find $C$ , cover $(x - 2)$ on the left side and substitute $x = 2$ into the remaining expression: $\frac{2 + 2}{2 - 3} = \frac{4}{- 1} = - 4$ .

Step 3: Calculate the Sum of Coefficients○ Expand

We have found the values of $A, B, C,$ and $D$ :

A = 2

B = 3

C = - 4

D = 5

Now, sum these values:

A + B + C + D = 2 + 3 + (- 4) + 5 = 5 - 4 + 5 = 1 + 5 = 6

The sum $A + B + C + D$ is 6.

💡 Teacher's Secret Hint

Double-check your arithmetic, especially with negative numbers, to avoid simple calculation errors.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 19 – AP-EAMCET 2026

Clue 1 of 3: Rational Functions

🧠 Full Solution Path