To calculate: The partial decomposition of the function \frac{-12x-

vadulgattp

vadulgattp

Answered question

2021-11-14

To calculate: The partial decomposition of the function 12x292x2+11x+15.

Answer & Explanation

Nancy Johnson

Nancy Johnson

Beginner2021-11-15Added 17 answers

Given Information:
The provided expression is 12x292x2+11x+15.
Formula used:
Decomposition of f(x)g(x) into Partial Fractions:
Consider a rational expression f(x)g(x), where f(x) and g(x) are polynomial with real coefficients, g(x)0, and the degree of f(x) is less than degree of g(x).
Step 1: Factor the denominator g(x) completely into linear factors of the form (ax+b)m and quadratic factors of the form (ax2+bx+c)n that are not further factorable over the integers.
Step 2: Set up the form of decomposition. That is, write the original rational expression f(x)g(x) as a sum of simpler fractions using these guidelines. Note that A1,A2,.Am,B1,B2,.,Bm and C1,C2,.Cm are constants.
Linear Factors of g (x):
For each linear factor of g(x), the partial fraction decomposition must include the sum:
A1(ax+b)1+A{(ax+b)2++Am(ax+b)m
Quadratic Factors of g(x):
For each quadratic factor of g(x), the partial fraction decomposition must include the sum:
B1x+C1(ax2+bx+c)1+B2x+C2(ax2+bx+c)2++Bnx+Cn(ax2+bx+c)n
Step 3: With the form of the partial fraction decomposition set up, multiply both sides of the equation by the Least Common Divisor to clear fractions.
Step 4: Use the equation from step 3, set up a system of linear equations by equating the constant terms and equating the coefficients of like powers of x.
Step 5: Solve the system of equations from step4 and substitute the solutions to the system into the partial fraction decomposition.
Calculation:
Consider the provided expression,
12x292x2+11x+15
Here, f(x)=12x29 and g(x)=2x2+11x+15
Factorize g(x):
g(x)=2x2+11x+15
=2x2+6x+5x+15
=2x(x+3)+5(x+3)
=(2x+5)(x+3)
So, the expression can be written as:

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