Factoring techniques
Factoring reverses multiplication, rewriting a polynomial as a product. It is the key step for solving equations, simplifying fractions, and finding where graphs cross the axis.
Greatest common factor
Always start by factoring out the greatest common factor (GCF) of all terms. \(6x^3 + 9x^2 - 15x = 3x(2x^2 + 3x - 5)\). Check by distributing back. Taking out a negative GCF is useful when the leading coefficient is negative: \(-x^2 + 4x = -x(x - 4)\).
Factoring by grouping
With four terms, group in pairs and factor each pair: \(x^3 + 2x^2 + 3x + 6 = x^2(x + 2) + 3(x + 2) = (x + 2)(x^2 + 3)\). Grouping works when the pairs share a common binomial factor.
Trinomials with leading coefficient 1
To factor \(x^2 + bx + c\), find two numbers that multiply to c and add to b. For \(x^2 - 2x - 15\), those numbers are -5 and 3, so it factors as \((x - 5)(x + 3)\). If c is positive, both numbers share the sign of b; if c is negative, the signs differ.
Trinomials with other leading coefficients
For \(ax^2 + bx + c\), the AC method finds two numbers that multiply to ac and add to b, then splits the middle term and groups. For \(6x^2 + 7x - 3\), ac = -18, and 9 and -2 work: \(6x^2 + 9x - 2x - 3 = 3x(2x + 3) - 1(2x + 3) = (2x + 3)(3x - 1)\).
Special patterns
Recognise a difference of squares, \(a^2 - b^2 = (a + b)(a - b)\), as in \(x^2 - 49 = (x + 7)(x - 7)\); perfect-square trinomials, \(x^2 + 10x + 25 = (x + 5)^2\); and sums and differences of cubes, \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) and \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). A sum of squares, such as \(x^2 + 9\), does not factor over the real numbers.
A factoring strategy
First take out the GCF. Then count terms: two terms suggest a difference of squares or cubes; three terms suggest a trinomial or perfect square; four terms suggest grouping. Finally, check whether any factor can be factored further, and verify by multiplying back.
Factoring completely
A polynomial is factored completely when no factor can be factored further. For example, \(2x^3 - 18x = 2x(x^2 - 9) = 2x(x + 3)(x - 3)\). Stopping at \(2x(x^2 - 9)\) is a common error.
Worked example
Factor \(3x^4 - 48\). The GCF is 3: \(3(x^4 - 16)\). That is a difference of squares: \(3(x^2 + 4)(x^2 - 4)\). The second factor is again a difference of squares: \(3(x^2 + 4)(x + 2)(x - 2)\). The sum of squares \(x^2 + 4\) does not factor further over the reals.
Key takeaways
Factoring rewrites a polynomial as a product and reverses multiplication. Start with the greatest common factor. Use grouping for four terms. Factor x squared + bx + c with two numbers multiplying to c and adding to b; use the AC method when the leading coefficient is not 1. Know the difference of squares, perfect-square trinomials, and sums and differences of cubes; sums of squares do not factor over the reals. Factor completely, and check by multiplying back.
Flashcard
Writing as a product
Check your understanding
Factor \(x^2 - 2x - 15\).