To express the given expression as partial fractions, we need to factor the denominator and determine the values of the unknown constants.
The given expression is:
StartFraction 8 x cubed minus 2 x squared minus 67 x minus 24 Over 2 x squared minus x minus 21 EndFraction
First, we factor the denominator by finding two numbers whose product is equal to the product of the coefficient of x squared term (2) and the constant term (-21), and whose sum is equal to the coefficient of the x term (-1). The factored form of the denominator is:
(2x + 3)(x - 7)
Now, we can express the given expression as the sum of partial fractions:
StartFraction 8 x cubed minus 2 x squared minus 67 x minus 24 Over (2x + 3)(x - 7) EndFraction
= StartFraction A Over 2x + 3 EndFraction + StartFraction B Over x - 7 EndFraction
To find the values of A and B, we can multiply both sides of the equation by the denominator and simplify:
8x^3 - 2x^2 - 67x - 24 = A(x - 7) + B(2x + 3)
Expanding the right side of the equation:
8x^3 - 2x^2 - 67x - 24 = Ax - 7A + 2Bx + 3B
Grouping like terms:
8x^3 - 2x^2 - 67x - 24 = (A + 2B)x + (-7A + 3B)
By comparing the coefficients of like terms on both sides, we can form a system of equations:
A + 2B = 8
-7A + 3B = -67
Solving this system of equations will give us the values of A and B.
Now, let's find A and B by solving the system of equations:
From the first equation, we can isolate A:
A = 8 - 2B
Substitute this value of A into the second equation:
-7(8 - 2B) + 3B = -67
Simplifying:
-56 + 14B + 3B = -67
17B = -67 + 56
17B = -11
B = -11/17
Substituting the value of B back into A = 8 - 2B:
A = 8 - 2(-11/17)
A = 8 + 22/17
A = (136 + 22)/17
A = 158/17
Therefore, the expression in partial fraction form is:
StartFraction 8 x cubed minus 2 x squared minus 67 x minus 24 Over 2 x squared minus x minus 21 EndFraction
= StartFraction 158 Over 17 x + ( -11 Over 17 ) Over 2x + 3 EndFraction + StartFraction -11 Over 17 x + ( 158 Over 17 ) Over x - 7 EndFraction
So, the correct expression representing the quotient and remainder, written as partial fractions, is:
A + StartFraction B Over x + 3 EndFraction + StartFraction C Over x - 7 EndFraction
where A = 158/17, B = -11/17, and C = 158/17.