Enter partial fraction expression \frac{(11-3x)}{(x^2+2x-3)} as (11-3x)/(x^2+2x-3)

1) Decompose the rational function (3x^2 + 5x + 2) / (x^2 – 4x + 4) into partial fractions.

2) Solve the partial fraction decomposition of (4x + 10) / (x^2 + 5x + 6) containing repeated linear factors.

3) Decompose the rational function (2x^2 + 3x + 1) / (x^3 – 4x^2 + 4x) into partial fractions using the method of solving partial fractions.

4) Determine the partial fraction decomposition of (x^2 + 3x + 2) / (x^2 – 2x – 3) with quadratic factors.

Partial fractions is a way to break down a complicated fraction into simpler parts.

Imagine you have a big piece of pizza, and you want to share it with your friends. But instead of giving them the whole pizza at once, you break it down into smaller slices to make it easier to share. That’s kind of like what we do with fractions in math!

Let’s say we have the fraction (2x + 3) / (x^2 + 5x + 6). This fraction can be broken down into two simpler fractions: A/x + B/(x+3), where A and B are numbers that we need to find.

To find the values of A and B, we can use a method called method of solving partial fractions. We can rewrite the original fraction as:

(2x + 3) / (x^2 + 5x + 6) = A/x + B/(x+3)

Now, we need to find the values of A and B by simplifying the equation. We can clear the fractions by multiplying both sides by the denominator (x^2 + 5x + 6):

2x + 3 = A(x+3) + B(x)

Expanding the right side of the equation gives us:

2x + 3 = Ax + 3A + Bx

Now we can combine like terms to solve for A and B. So, if we match coefficients of x on both sides of the equation:

2 = A + B (coefficient of x terms)

And also by matching the constant terms:

3 = 3A (constant terms)

From these two equations, we can solve for A and B.

Now, let’s find the values of A and B for our example fraction (2x + 3) / (x^2 + 5x + 6):

With A = 1 and B = 1, we will substitute those into the original equation:

(2x + 3) / (x^2 + 5x + 6) = 1/x + 1/(x+3)

By using the method of solving partial fractions, we can break down a complicated fraction into simpler parts, making it easier to solve and understand.

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