FactorisationFind the structure.
Build the skill.

1. 1

   Find the greatest common factor of the coefficients. For fractions, use a common denominator; for decimals, scale to integers and undo the scale after factoring.

2. 2

   For a shared variable, choose its lowest power. A constant has no variable factor.

3. 3

   Divide BOTH terms by the factor. Put the results in a bracket and expand to check.

A factor must divide every term. In 6x + 9, you may take out 3, but not x.

1. 1

   Reorder terms if it helps, and combine like terms first.

2. 2

   Check every coefficient and every variable power before choosing a factor.

3. 3

   A constant blocks a variable factor. If the common factor is 1, say so.

More terms do not automatically mean a harder method. Always inspect all the terms.

1. 1

   Find the coefficient factor as before.

2. 2

   Inspect each variable separately and keep its lowest shared power.

3. 3

   A variable present in only one term stays inside the bracket.

Use the minimum exponent for EACH variable. A missing variable has exponent zero.

1. 1

   Scan all terms, including constants and shuffled terms.

2. 2

   Build the common factor variable by variable; include only variables in every term.

3. 3

   Leave any unfactorable remainder in the bracket and check the full product.

A factor shared by most terms is not a common factor of the whole expression.

1. 1

   Try pairs that share useful factors; terms may need reordering.

2. 2

   Factor each group so that the SAME bracket appears.

3. 3

   Take that bracket out as a factor. If it does not match, try another grouping.

Factoring two pairs is only the intermediate step. Finish with a product.

1. 1

   Treat a repeated bracket as one object.

2. 2

   Remember b − a = −(a − b). Change every sign when taking out a negative.

3. 3

   Use the matching bracket as the common factor, then check the remaining factors.

Reversing a subtraction changes its sign. (x − 2) and (2 − x) are opposites.

1. 1

   Remove any common factor first.

2. 2

   For x² + bx + c, find two numbers whose product is c and sum is b.

3. 3

   Write (x + p)(x + q). If p = q, you have a repeated factor.

The pair must satisfy both conditions. A positive constant does not always mean positive factors.

1. 1

   Remove a common factor; clear fractional coefficients if helpful.

2. 2

   Find two numbers with product ac and sum b.

3. 3

   Split bx using those numbers, group, and factor the matching brackets.

Use ac, not just c, when the coefficient of x² is not 1.

1. 1

   Remove the common factor first.

2. 2

   Check that there are two square terms separated by subtraction.

3. 3

   Use A² − B² = (A − B)(A + B). Check whether a factor can be factored again.

A sum of squares does not use this identity. x² + 9 is not (x − 3)(x + 3).

1. 1

   Start with a common factor whenever there is one.

2. 2

   Recognise A² ± 2AB + B² = (A ± B)², or choose another learned method.

3. 3

   Inspect every bracket and continue until its rational polynomial factors cannot be reduced further.

2x(x² − 4) is correct but incomplete when the instruction says factorise completely.

1. 1

   Remove the common factor and identify exact cubes.

2. 2

   Use A³ − B³ = (A − B)(A² + AB + B²), or A³ + B³ = (A + B)(A² − AB + B²).

3. 3

   Check the signs by expansion; inspect the quadratic factor for any further rational factors.

A sum of cubes factors over the rationals; a sum of two squares generally does not.

1. 1

   For polynomial P, if P(r) = 0, then x − r is a factor.

2. 2

   Use polynomial or synthetic division to remove the known factor.

3. 3

   Factor the remaining quadratic and check the full product.

A root r gives factor x − r. A root of −2 gives factor x + 2.

1. 1

   Look for a repeated structure such as x². Temporarily call it u.

2. 2

   Factor in u, substitute back, and inspect each new bracket.

3. 3

   State the number system. A polynomial may be irreducible over the rationals but factor over the reals.

Substitution can reveal a method, but it is not the final answer. Replace u and factor again.