Methods of factorization of a polynomial: factoring out a common factor, abbreviated multiplication formulas, grouping

A monomial is a product of numbers and variables raised to natural powers. A polynomial is a sum of monomials. An identity transformation that transforms a polynomial into a product of several factors is called factorization. There are three main ways to factor polynomials: factoring out a common factor, abbreviated multiplication formulas, and the grouping method.

Abbreviated multiplication formulas

a² – b² = (a - b)(a + b)
a³ – b³ = (a - b)( a² + ab + b²)
a³ + b³ = (a + b)( a² - ab + b²)

Examples of the application of abbreviated multiplication formulas for factorization

1)   a⁴ – 16 = (a²–4)(a²+4) = (a–2)(a+2)(a²+4).
2)   c⁶ – 1 = (³– 1)(³+1) = (–1)(²++1)(+1)(²–+1).
3)   a⁸ – 1 = (a⁴–1)(a⁴+1) = (a²–1)(a²+1)(a⁴+1) = (a–1)(a+1)(a²+1)(a⁴+1).

Examples of a combination of factoring out a common factor and grouping terms

1)   10ay – bx + 2ax – 5by = (10ay–5by) + (2ax–bx) = 5y(2a–b) + x(2a–b) = (2a–b)(5y+x).
2)   16ab² - 10c³ + 32ac² - 5b²c = (16ab²+32ac²) – (5b²c+10c³) = 16a(b²+2c²) – 5c(b²+2c²) = (b²+2c²)(16a–5c).

Examples of combinations of factoring, grouping terms, and abbreviated multiplication formulas for factoring polynomials

1)   y³ + 16 – 4y – 4y² = (y³–4y) + (16-4y²) = (y³–4y) – (4y²–16) = y(y²–4) – 4(y²–4) = (y²–4)(y-4) = (y–2)(y+2)(y-4).

2)   (a – b)³ – a + b = (a–b)³ – (a–b) = (a–b)((a–b)²–1) = (a–b)(a–b-1)(a–b+1).

3)   x² – 6xy – 49 + 9y² = (x²–6xy+9y²) – 49 = (x–3y)² – 49 = (x–3y–7)(x–3y+7).

4)   c² + 2c – d² – 2d = (c²–d²) + (2c–2d) = (c–d)(c+d) + 2(c–d) = (c–d)(c+d+2).

Examples of non-standard factorizations of polynomials

One or more terms are represented as a sum or difference, after which grouping or abbreviated multiplication formulas can be applied.

1)   y²– 14y + 40 = y²–14y+49–9 = (y²–14y+49) – 9 = (y–7)² – 3² = (y–7–3)(y–7+3) = (y–10)(y–4).

2)   x²+ 7x + 12 = x²+3x+4x+12 = (x²+3x) + (4x+12) = x(x+3) + 4(x+3) = (x+3)(x+4).

3)   x² + 8x +7 = x²+7x+x+7 = (x²+7x) + (x+7) = x(x+7) + (x+7) = (x+7)(x+1).

4)   x² + x – 12 = x²+4x–3x–12 = (x²+4x) – (3x+12) = x(x+4) – 3(x+4) = (x+4)(x–3).

5)   x² - 10x + 24 = x²-2*5 x+25–1 = (x²-2*5x+25) – 1 = (x–5)² – 1 = (x– 5–1)(x–5+1) = (x–6)(x–4).

6)   x² - 13x + 40 = x²-10x–3x+25+15 = (x²-10x+25) – (3x–15) = (x–5)² – 3(x–5) = (x–5)(x–5–3) = (x–5)(x–8).

7)   x² + 15x + 54 = x²+(12x+3x) + (36+18) = (x²+12x+36) + (3x+18) = (x+6)² + 3(x+6) = (x+6)(x+6+3) = (x+6)(x+9).

8)   x⁴ + 3x² + 4 = x⁴+(4x²–x²) + 4 = (x⁴+4x²+4) – x² = (x²+2)²–x² = (x²+2–x)(x²+2+x) = (x²–x+2)(x²+x+2).

9)   x⁴ + x² + 1 = x⁴+(2x²–x²) + 1 = (x⁴+2x²+1) – x² = (x²+1)² – x² = (x²+1–x)(x²+1+x) =(x²– x+1)(x²+x+1).

10)   x⁴ + 4 = x⁴ + 4 + 4x² – 4x² = (x⁴+4x²+4) – 4x² = (x²+2)² – 4x² = (x²+2–2x)(x²+2+2x) = (x²–2x+2)(x²+2x+2).