Question

Найти a и b из тождества (2x+3)/(x+2)*(x+1) = a/(x+1) + b/(x+2) Честно говоря, не совсем понимаю, что требуется....

Answer 1

(2x+3)/(x+2)(x+1) = a/(x+1) +b/(x+2) <=>
(2x+3)/(x+2)(x+1)=[a(x+2)+b(x+1)]/(x+1)(x+2) <=> 
2x+3 = a(x+2) +b(x+1) <=>
2x+3 = ax +2a +bx +b <=>

{ax+bx=2x => a+b=2 <=> b=2-a
{2a+b=3

{b=2-a
{2a+2-a=3 <=> a=1

b=2-1=1

Answer 2

Solving the Equation (2x+3)/(x+2)*(x+1) = a/(x+1) + b/(x+2)

To solve the given equation (2x+3)/(x+2)*(x+1) = a/(x+1) + b/(x+2), we can use the method of partial fraction decomposition to express the left-hand side of the equation as the sum of two simpler fractions.

Partial Fraction Decomposition

The process of partial fraction decomposition involves expressing a rational function as the sum of simpler fractions. In this case, we want to express the left-hand side of the equation in the form a/(x+1) + b/(x+2).

By equating the numerators of the fractions on both sides of the equation, we can find the values of a and b.

Finding the Values of a and b

To find the values of a and b, we can start by multiplying both sides of the equation by the common denominator (x+1)(x+2) to clear the fractions. This will allow us to compare the numerators on both sides and solve for a and b.

After multiplying and simplifying, we can compare the coefficients of like terms on both sides of the equation to determine the values of a and b.

Conclusion

By following the process of partial fraction decomposition and comparing the coefficients, we can find the values of a and b that satisfy the given equation.

Let's proceed with the calculations to find the specific values of a and b.