Polynomials & Algebraic Fractions – Revision Flashcards

Question-and-answer flashcards covering expansion, factorisation, and operations with algebraic fractions.

How do you expand a single or double bracket?

Apply the distributive law: (a + b)(c + d) = ac + ad + bc + bd.

What is the correct expansion of (a + b)^2 ?

(a + b)^2 = a^2 + 2ab + b^2 (not a^2 + b^2).

What is the identity for the difference of two squares?

a^2 − b^2 = (a − b)(a + b).

Expand and simplify (2x + 3)(6x^2 + 6x + 3).

(2x + 3)(6x^2 + 6x + 3) = 12x^3 + 30x^2 + 24x + 9.

How do you simplify an expression using common factors?

Identify any common number, variable, or bracket and factor it out. Example: x(1 + x − 2) + (1 + x)(1 − x) = (1 + x)(−1).

How do you factorise a quadratic when a = 1?

Find two numbers that multiply to c and add to b, then write two brackets. Example: x^2 + 4x − 21 = (x + 7)(x − 3).

How do you factorise a quadratic when a ≠ 1?

1) Multiply a × c. 2) Find two numbers that multiply to this product and add to b. 3) Split the middle term and factorise in pairs. Example: 3x^2 + 4x − 15 = (3x − 5)(x + 3).

What is the first step when simplifying an algebraic fraction?

Factorise both numerator and denominator, then cancel any common factors.

What do you do if there is a fraction within the numerator or denominator of an algebraic fraction?

Multiply the entire numerator and denominator by the same factor to clear the internal fraction. Example: (2 + 1/x)/(x + 1) → multiply top and bottom by x.

How do you divide algebraic fractions?

Take the reciprocal (flip) of the second fraction and multiply.

What are the steps to add or subtract algebraic fractions?

1) Find a common denominator. 2) Rewrite each fraction with that denominator. 3) Combine the numerators. 4) Simplify if possible; e.g., (2y)/(x(x + 3)) + 1/(y(x + 3)) − x/y = (2y^2 + x − x^2y − 3xy)/(xy(x + 3)).