Partial fractions Flashcards

1
Q

What is a proper rational expression?

A

Where the degree of the top is less than the degree of the bottom.

For partial fractions, the denominator must always be of higher order then the numberator.

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2
Q

What is an irreducible quadratic?

A

A quadratic that cannot be further factorised.

e. g. a2 + b2 – has no real factors
e. g. x2 + 4

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3
Q

Describe the Cover up method.

A

1) Factorise the denominator.
2) Each factor gives rise to an expression.

So if there are 3 factors, there will be 3 terms.

3) Multiply by the denominator
4) Choose x-values that simplify the problem (usually zeros of the factors).
5) Solve simaltaneous equations.

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4
Q

Describe another method (not the Cover up) of solving partial fractions?

A

Equating the powers of x.

1) Once the equation is in the form of x + c = A(x - a) + B(x - b), expand and group like terms.
2) Equate the coefficients of powers of x.
e. g. 5x+ c = (A+B)x + -aA + -bB

coefficient of x1 = 5, therefore A+B = 5.

coefficient of x0 = 1, therefore -aA + -bB = c

3) Now solve the simaltaneous equations.

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5
Q

Explain how to deal with repeated factors.

A

When a factor is repeated, it adds an extra possibility when solving.

Once you have identified this and written the equation out to include the repeated factor, you can continue as normal.

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6
Q

What if the denominator contains a quadratic?

A

We have to consider that the numerator may contain an x term.

As long as the degree of the numerator is lower than the degree of the denominator, this is still a proper fraction.

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