Define rational expression .

A rational expression is an algebraic fraction: one algebraic expression divided by another, usually two polynomials. The name comes from ratio , in just the same way that a rational number is a ratio of two integers.

What are the two moves that simplify a rational expression?

Factorise the numerator and the denominator, then cancel any factor common to both. Nothing can be cancelled until both parts are written as products, which is why the factorising has to come first.

After cancelling, what should you check about the answer?

Whether it is still top-heavy , with the numerator's degree at least as large as the denominator's. A top-heavy answer usually needs simplifying further before it can be used for anything else.

When dividing by a quadratic, what do you divide the leading term by?

The squared term of the divisor, rather than the whole of it. That gives the leading term of the quotient, which is then multiplied by the divisor and subtracted in the usual way.

International A Level (IAL)MathsEdexcelPure 3FlashcardsAlgebra & FunctionsRational Expressions

Rational Expressions (Edexcel International A Level (IAL) Maths: Pure 3): Flashcards

Exam code: YMA01

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Define rational expression.

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Define rational expression.
A rational expression is an algebraic fraction: one algebraic expression divided by another, usually two polynomials.
The name comes from ratio, in just the same way that a rational number is a ratio of two integers.
What are the two moves that simplify a rational expression?
Factorise the numerator and the denominator, then cancel any factor common to both.
Nothing can be cancelled until both parts are written as products, which is why the factorising has to come first.
True or False?
In $\frac{x + 3}{x + 6}$ the two $x$ terms cancel, leaving $\frac{3}{6}$ .
False.
Only factors can be cancelled, and here each $x$ is a term added on rather than something multiplying the whole of the numerator or denominator.
Cancelling is really dividing top and bottom by the same thing, which only works on something that multiplies all of it.
Complete the simplification by cancelling:
$\frac{x^{3} - 7 x + 6}{x^{2} + 2 x - 3} = \frac{(x - 1) (x + 3) (x - 2)}{(x + 3) (x - 1)} =______$
The completed simplification is:
$\frac{x^{3} - 7 x + 6}{x^{2} + 2 x - 3} = x - 2$
Both $(x - 1)$ and $(x + 3)$ appear top and bottom, so both disappear and a single factor is left.
After cancelling, what should you check about the answer?
Whether it is still top-heavy, with the numerator's degree at least as large as the denominator's.
A top-heavy answer usually needs simplifying further before it can be used for anything else.
Define improper algebraic fraction.
An improper algebraic fraction is one whose numerator has a degree greater than or equal to that of the denominator.
So $\frac{x^{3} + 2 x^{2} - x + 4}{x - 5}$ is improper, being degree $3$ over degree $1$ .
True or False?
$\frac{x}{x + 1}$ is an improper algebraic fraction.
True.
Both the numerator and the denominator have degree $1$ , so the top is not smaller than the bottom.
It rewrites as $1 - \frac{1}{x + 1}$ , which is easy to overlook because there is no long division to carry out.
Complete the form an improper fraction takes when it is divided by a linear expression:
$\frac{p (x)}{a x + b} \equiv q (x) + \frac{______}{______}$
The completed form is:
$\frac{p (x)}{a x + b} \equiv q (x) + \frac{r}{a x + b}$
The remainder is written back over the same divisor it came from.
What numerical process is rewriting an improper algebraic fraction like?
Turning a top-heavy number into a mixed number.
Just as $\frac{17}{5} = 3 \frac{2}{5}$ because $17 \div 5$ is $3$ remainder $2$ , the algebraic version splits into a whole part plus whatever is left over.
What form does $\frac{f (x)}{a x^{2} + b x + c}$ take once rewritten?
$q (x) + \frac{r (x)}{a x^{2} + b x + c}$ , with the remainder written back over the quadratic.
If $f (x)$ has degree $n$ , then the quotient $q (x)$ has degree $n - 2$ .
True or False?
When a polynomial is divided by a quadratic, the remainder is always a number.
False.
The remainder only has to be of lower degree than the divisor, so against a quadratic it may be linear, such as $3 x - 1$ .
It is a linear divisor that forces the remainder all the way down to a single number.
How do you rewrite $\frac{x^{3} + 3 x^{2} - 4 x + 2}{x - 1}$ as a polynomial plus a fraction?
Carry out the algebraic division, then write the remainder back over the divisor.
Here the quotient is $x^{2} + 4 x$ and the remainder is $2$ , giving $x^{2} + 4 x + \frac{2}{x - 1}$ .
When dividing by a quadratic, what do you divide the leading term by?
The squared term of the divisor, rather than the whole of it.
That gives the leading term of the quotient, which is then multiplied by the divisor and subtracted in the usual way.

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