MathCrave AI partial fractions solver uses artificial intelligence techniques to solve problems related to partial fraction decomposition. It takes an algebraic fraction that cannot be simplified further and breaks it down into simpler fractions. The AI component of the solver allows it to analyze the given fraction, determine the appropriate decomposition method, and provide a step-by-step solution to solve the following problems.
Partial fractions a fraction containing linear factors in the denominator
Partial fractions a fraction containing repeated linear factors in the denominator
Partial fractions a fraction containing quadratic factors in the denominator
If you are resolving a partial fraction containing linear factors
Resolve (11 −3x) / (x^2 + 2x − 3) into partial fractions
Resolve (x^2 + 1)/ ( x^2 − 3x + 2) into partial fractions
If you are resolving a partial fraction containing repeated linear factors
Resolve (2x + 3) / (x − 2)^2 into partial fractions
Resolve (3x^2 + 16x + 15) / (x + 3)^3 into partial fractions
To resolve a fraction into partial fractions, certain conditions need to be met.
The degree of the denominator must be greater than or equal to the degree of the numerator.
If the numerator's degree is equal to the denominator's degree, it is necessary to perform polynomial division first.
The roots of the denominator must be distinct.
When the denominator of a fraction contains linear factors, the partial fraction decomposition involves expressing the fraction as the sum of simpler fractions. Each of these simpler fractions corresponds to one of the linear factors in the denominator. Suppose the denominator has factors (x - a)(x - b)(x - c), then the partial fraction decomposition will have the form:
A/(x - a) + B/(x - b) + C/(x - c).
In the case of repeated linear factors in the denominator, the partial fraction decomposition requires additional terms. Suppose the denominator has repeated factors (x - a)^n, then the partial fraction decomposition will have the following form:
A_1/(x - a) + A_2/(x - a)^2 + ... + A_n/(x - a)^n.
When the denominator of a fraction contains quadratic factors, the partial fraction decomposition involves expressing the fraction as the sum of simpler fractions. Each of these simpler fractions corresponds to one of the quadratic factors in the denominator. Suppose the denominator has factors (ax^2 + bx + c)(dx^2 + ex + f), then the partial fraction decomposition will have the form:
(Ax + B)/(ax^2 + bx + c) + (Cx + D)/(dx^2 + ex + f).
