To determine To calculate: The partial decomposition of the functi

ukelangf0

ukelangf0

Answered question

2021-11-16

To determine
To calculate: The partial decomposition of the function 6x+8x2+4x+4.

Answer & Explanation

Ched1950

Ched1950

Beginner2021-11-17Added 21 answers

Calculation:
Consider the provided expression,
Here, f(x)=6x+8 and g(x)=x2+4x+4
Factorize g(x):
g(x)=x2+4x+4
=x2+2x+2x+4
=x(x+2)+2(x+2)
=(x+2)2
So, the expression can be written as,
6x+8x2+4x+4=6x+8(x+2)2
Here, in g (x), there is one linear factor with square power. So, the expression can be decomposed as:
6x+8(x+2)2=Ax+2+B(x+2)2(1)
Here, Least Common Divisor (x+2)2
Multiply both sides of (1) by the Least Common Divisor to clear fractions:
(x+2)2[6x+8(x+2)2]=(x+2)2[Ax+2+B(x+2)2]
Simplify to obtain:
6x+8=A(x+2)+B
6x+8=Ax+2A+B
6x+8=(A)x+(2A+B)
Compare the coefficients of x and constant terms:
A=6 nd2A+B=nd2A+B=8
Substitute 6 for A in equation 2A + B = 8 and simplify for B:
2(6)+B=8
12+b=8
B=4
Substitute the obtained values of A and B in equation (1):
6x+8(x+2)2=6x+2+4(x+2)2
Therefore, the partial fraction decomposition for 6x+8(x+2)2s6x+2+4(x+2)2s6x+2+4(x+2)2

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