Is this right?
Decompose into partial fractions.
(2x^2-24x+35)/(x^2+2x-7)(x-2)=
(Ax+B)/(x^2+2x-7)+C/(x-2)
=Ax^2+Bx-2Ax-2B+Cx^2+2xC-7C
=(A+C)x^2+Bx+2(Ax+Cx-B)-7C
A+C=2
2Ax+Bx+2Cx=24
-2B-7C=35
A=2-C
B=(35-7C)/(-2)
My answers are
A=2-C
B=(35-7C)/(-2)
I agree to here:
Ax^2+Bx-2Ax-2B+Cx^2+2xC-7C
but then
(A+C)x^2 + (B-2A+2C) x +(-2B-7C)
so
A+C=2 so C=(2-A)
B-2A+2C=-24
-2B-7C = 35
solve for A and B using (2-A) for C
B-2A+2(2-A) = -24
2B+7(2-A) = -35
B - 4 A = -28
2B - 7A = -49
2 B - 8 A = -56
2 B - 7A = -49
subtract
-A = -7
so
A = 7
then B = 4(7) - 28 = 0
C = 2-A = -5
7, 0 , -5 indeed work
To decompose the given rational expression into partial fractions, follow these steps:
Step 1: Factor the denominator
The denominator of the rational expression is a quadratic expression. Factor it to find its roots:
x^2 + 2x - 7 = (x - 1)(x + 7)
Step 2: Write the partial fraction form
Since the denominator has two distinct linear factors, the partial fraction form will have two fractions:
(2x^2 - 24x + 35)/((x - 1)(x + 7)(x - 2)) = (A/(x - 1)) + (B/(x + 7)) + (C/(x - 2))
Step 3: Clear the fractions
Multiply both sides of the equation by the denominator to clear the fractions:
(2x^2 - 24x + 35) = A(x + 7)(x - 2) + B(x - 1)(x - 2) + C(x - 1)(x + 7)
Step 4: Expand and simplify the equation
Expand the right side of the equation and simplify:
2x^2 - 24x + 35 = A(x^2 - 2x + 7x - 14) + B(x^2 - 3x + 2) + C(x^2 + 6x - 7x - 7)
2x^2 - 24x + 35 = A(x^2 + 5x - 14) + B(x^2 - 3x + 2) + C(x^2 - x - 7)
Step 5: Group like terms
Group the terms with the same powers of x:
(2x^2 - 24x + 35) = (A + B + C)x^2 + (5A - 3B - C)x + (-14A + 2B - 7C)
Step 6: Set up the system of equations
The coefficients of the terms on the left side of the equation should be equal to the coefficients of the corresponding terms on the right side. Setting them equal to each other, we have:
A + B + C = 2
5A - 3B - C = -24
-14A + 2B - 7C = 35
We now have a system of three equations to solve for A, B, and C.
Step 7: Solve the system of equations
You can solve this system of equations using various methods, such as substitution or elimination. In this case, let's solve it using substitution.
From equation 1, A = 2 - B - C
Substitute A = 2 - B - C into equations 2 and 3 to eliminate A:
5(2 - B - C) - 3B - C = -24
-14(2 - B - C) + 2B - 7C = 35
Simplify the equations:
10 - 5B - 5C - 3B - C = -24
-28 + 14B + 14C + 2B - 7C = 35
Combine like terms:
-8B - 6C = -34
-12B + 7C = 63
Multiply the first equation by 3 and the second equation by 2:
-24B - 18C = -102
-24B + 14C = 126
Subtract the first equation from the second equation:
32C = 228
Divide both sides by 32:
C = 228/32
C = 7.125
Now substitute C = 7.125 into the first equation to solve for B:
-8B - 6(7.125) = -34
-8B - 42.75 = -34
-8B = -34 + 42.75
-8B = 8.75
B = 8.75/-8
B = -1.09375
Lastly, substitute the values of B and C into the equation A = 2 - B - C to solve for A:
A = 2 - (-1.09375) - 7.125
A = 2 + 1.09375 - 7.125
A = -4.03125
Therefore, the partial fraction decomposition of the given expression is:
(2x^2 - 24x + 35)/((x - 1)(x + 7)(x - 2)) = (-4.03125/(x - 1)) + (-1.09375/(x + 7)) + (7.125/(x - 2))