Factoring patterns

GCF first

To factor means to rewrite as a product. Take out the GCF: the largest number and the lowest power of each shared variable.

Example: \(6x^3 - 9x^2 = 3x^2(2x - 3)\); \(\;-4x^2 + 8x = -4x(x - 2)\).

Example: A binomial GCF: \(x(x + 5) + 2(x + 5) = (x + 5)(x + 2)\).

Grouping (4 terms)

Group the terms in pairs, take the GCF out of each pair, then take out the shared binomial.

Example: \(x^3 + 3x^2 + 2x + 6 = x^2(x + 3) + 2(x + 3) = (x + 3)(x^2 + 2)\).

\(x^2 + bx + c\)

Find two numbers that multiply to \(c\) and add to \(b\). Signs: \(c \gt 0\) means same signs (the sign of \(b\)); \(c \lt 0\) means different signs.

Example: \(x^2 + 7x + 12 = (x + 3)(x + 4)\); \(\;x^2 - 2x - 15 = (x - 5)(x + 3)\); \(\;x^2 + 5xy + 6y^2 = (x + 2y)(x + 3y)\).

No pair works? It is a prime polynomial, like \(x^2 + 3x + 5\).

\(ax^2 + bx + c\): the ac method

GCF first. Find two numbers that multiply to \(ac\) and add to \(b\). Split the middle term with them, then factor by grouping.

Example: \(6x^2 + 7x - 3\): \(ac = -18\); \(9\) and \(-2\). \(6x^2 + 9x - 2x - 3 = 3x(2x + 3) - 1(2x + 3) = (2x + 3)(3x - 1)\).

Difference of squares

Difference of squares: \(a^2 - b^2 = (a + b)(a - b)\).

Example: \(9x^2 - 25 = (3x + 5)(3x - 5)\).

Watch out: a sum of squares, like \(x^2 + 9\), is prime.

Perfect square trinomials

Perfect square trinomial: \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 - 2ab + b^2 = (a - b)^2\). Test: first and last terms are squares and the middle is \(2ab\).

Example: \(x^2 - 10x + 25 = (x - 5)^2\); \(\;4x^2 + 12x + 9 = (2x + 3)^2\).

Sum and difference of cubes

Cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Signs: Same, Opposite, Always Positive. Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Example: \(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\); \(\;8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1)\).

Factor completely

Keep going until no factor can be factored again.

Example: \(2x^3 - 18x = 2x(x^2 - 9) = 2x(x + 3)(x - 3)\).

Example: \(x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)\).

Solving by factoring

1. Get standard form (everything on one side, 0 on the other). 2. Factor. 3. Use the zero product property: set each factor equal to 0. 4. Solve each.

Example: \(x^2 - 5x = -6\): \(x^2 - 5x + 6 = 0\), \((x - 2)(x - 3) = 0\), so \(x = 2\) or \(x = 3\).

Example: \(3x^2 = 12x\): \(3x^2 - 12x = 0\), \(3x(x - 4) = 0\), so \(x = 0\) or \(x = 4\).

Watch out: never divide both sides by \(x\); you would lose \(x = 0\).

Factoring word problems

Write the equation, solve by factoring, then throw out solutions that make no sense (like a negative length).

Example: A rectangle is 3 ft longer than wide with area 40 ft²: \(w(w + 3) = 40\), \((w + 8)(w - 5) = 0\). Width 5 ft, length 8 ft.