Simplify and operate on algebraic fractions

Which value of x makes this fraction undefined?

The original fraction has numerator x times the sum of x and 2 and denominator x. Cancelling the common factor leaves the sum of x and 2, but zero remains excluded. These bars show denominators, not the values of the fractions. Move x to zero and compare

At x = 0, what is true?

Fraction operations and their domains

This is extension towards Level 10 algebraic fractions. The same fraction rules apply to algebraic expressions, with the extra requirement to keep every original excluded value

1. 1Record restrictions first Every original denominator must be nonzero. When dividing, the divisor must also be nonzero
2. 2Cancel whole factors Factorise where needed. Divide a complete numerator and denominator by the same nonzero factor; do not cancel addends
3. 3Multiply or use a reciprocal Multiply numerators and denominators. For division, reverse only the divisor before multiplying
4. 4Use a common denominator For addition and subtraction, convert both fractions to the same denominator. Subtract the entire second numerator
5. 5Keep the original domain A cancelled factor still excludes its original zero. An equivalent simplified rule is used only at the original allowed inputs

Which simplification preserves the original restriction?

Write this as a polynomial over denominator 1. What is the coefficient of x?

Factorise the numerator and cancel x + 3. What is the constant in the reduced polynomial?

Which restriction must remain?

Which result follows after multiplying and cancelling the whole factor x?

The original product has denominators x and x − 1. What is the smaller excluded value?

After reversing only the divisor, which result is correct?

Which extra value is excluded because the divisor is zero?

Which set contains exactly the original excluded values?

The denominators match. What is the coefficient of x in the combined numerator?

What is the constant in that combined numerator?

Subtract the entire second numerator. What is the constant in the result numerator?

Which fraction gives the correct sum?

The denominators are x − 1 and (x − 1)². What power of x − 1 is required in the lowest common denominator?

Read the coefficient-table convention

A fraction can be scaled in many equivalent ways. These practice tables ask for an expanded monic denominator: its leading coefficient is 1. Divide every numerator and denominator coefficient by that same leading coefficient

1. 1A defined normal form Divide the numerator and denominator polynomials by three
2. 2Exact fractions are allowed The numerator then has x coefficient two-thirds and constant four-thirds; the denominator has x coefficient one and constant negative two
3. 3Include zero coefficients For a denominator x, enter leading coefficient 1 and constant 0.
4. 4State restrictions separately Normalising coefficients does not change the excluded values. Here x ≠ 2.

In the monic-denominator form, what is the coefficient of x in the numerator?

What is the numerator constant in the same monic-denominator form?

When the denominator is scaled to have leading coefficient 1, what is its constant term?

Which restriction remains?

Which answer retains the domain even though the numerator cancels to zero?