Using the cover up method, Solve the partial fraction 3X^2+3X-2/(X^2-1)(X+3)=P+Q/(X-1)+R/(X+1)
To solve the partial fraction decomposition using the cover-up method, we start by rewriting the denominator:
X^2 - 1 = (X - 1)(X + 1)(X + 3)
Therefore, we can rewrite the expression as:
(3X^2 + 3X - 2) / ((X - 1)(X + 1)(X + 3)) = P / (X - 1) + Q / (X + 1) + R / (X + 3)
To determine the values of P, Q, and R, we will use the cover-up method.
Cover-up method:
1. For P: Cover up (X - 1) and evaluate the expression at X = 1.
(3(1)^2 + 3(1) - 2) / ((1 - 1)(1 + 1)(1 + 3)) = P / (1 - 1)
(3 + 3 - 2) / (0 * 2 * 4) = P / 0
4 / 0 = P, which means P is undefined.
2. For Q: Cover up (X + 1) and evaluate the expression at X = -1.
(3(-1)^2 + 3(-1) - 2) / ((-1 - 1)(-1 + 1)(-1 + 3)) = Q / (-1 + 1)
(3 - 3 - 2) / (-2 * 0 * 2) = Q / 0
-2 / 0 = Q, which means Q is undefined.
3. For R: Cover up (X + 3) and evaluate the expression at X = -3.
(3(-3)^2 + 3(-3) - 2) / ((-3 - 1)(-3 + 1)(-3 + 3)) = R / (-3 + 3)
(3 * 9 - 9 - 2) / (-4 * -2 * 0) = R / 0
14 / 0 = R, which means R is undefined.
Since P, Q, and R are undefined, the partial fraction decomposition of the given expression cannot be solved using the cover-up method.