Express the following as a sum of partial fractions 5x^2+17x+15\(x +2)^2(x+1)

Question

Partial fraction

Answer 1

When simplifying using addition and subtraction, combine “like terms” by keeping the "like term" and adding or subtracting the numerical coefficients.

 

Examples:

3x + 4x = 7x 13xy – 9xy = 4xy 12x3y2 - 5x3y2 = 7x3y2

 

Why can’t you simplify?

4x3 + 4y3 11x2 – 7x 6x3y + 5xy3

 

 

Simplify:

 

1) 7x + 5 – 3x 2) 6w2 + 11w + 8w2 – 15w 3) (6x + 4) + (15 – 7x)

 

 

4) (12x – 5) – (7x – 11) 5) (2x2 - 3x + 7) – (-3x2 + 4x – 7) 6) 11a2b – 12ab2

 

Answer 2

Let's assume the answer is of the form A/(x+2)+B/(x+2)+C/(x+1). This reduces to a single fraction, with numerator:

A(x^2+3x+2)+B(x^2+3x+2)+C(x^2+4x+4)=

(A+B+C)x^2+x(3A+3B+4C)+2A+2B+4C. This must match 5x^2+17x+15:

A+B+C=5; 3A+3B+4C=17; 2A+2B+4C=15.

Let D=A+B: D+C=5; 3D+4C=17; 2D+4C=15: subtracting these last two equations: D=2; so C=3, but 3D+4C=18 which is not consistent with 3D+4C=17.

The answer may be in the form: (Ax+B)/(x^2+3x+2)+C/(x+2).

(Ax+B)(x+2)+C(x^2+3x+2)=x^2(A+C)+x(2A+B+3C)+2B+2C.

A+C=5; 2A+B+3C=17; 2B+2C=15. A=5-C, so 2(5-C)+B+3C=17, B+C=7, so 2(B+C)=14 but it should be 15!

Finally assume: (Ax+B)/(x^2+4x+4)+C/(x+1).

(Ax+B)(x+1)+C(x^2+4x+4)=x^2+17x+15.

A+C=5; A+B+4C=17; B+4C=15. A=2; C=3; B=3 fits all.

Answer: (2x+3)/((x+2)^2+3/(x+1).