Short multiplication formulas [Polynomial identities]

Square of sum and difference, difference of squares, cube of sum and difference, sum and difference of cubes formulas.

\begin{align} \displaystyle (a + b)^2 &= a^2 + 2ab+b^2 \\ \displaystyle (a - b)^2 &= a^2 - 2ab+b^2 \\ \displaystyle (a+b)(a-b) &= a^2-b^2 \\ \displaystyle (a + b)^3 &= a^3 + 3a^2b+3ab^2 + b^3 \\ \displaystyle (a - b)^3 &= a^3 - 3a^2b+3ab^2 - b^3 \\ \displaystyle (a - b)(a^2 + ab+b^2) &= a^3 - b^3 \\ \displaystyle (a + b)(a^2 - ab+b^2) &= a^3 + b^3 \end{align}

Solved examples

Simplify the expression $(2x-5)^2$

$$(2x-5)^2=(2x)^2-2\cdot 2x\cdot 5 + 5^2=4x^2-20x+25$$

Simplify the expression $(5x-4)(5x+4)$

$$(5x-4)(5x+4)=(5x)^2-4^2=25x^2-16$$

Simplify the expression $(3x-2)^3$

$$(3x-2)^3=(3x)^3-3\cdot (3x)^2\cdot 2 + 3 \cdot 3x \cdot 2^2 - 2^3=27x^3-54x^2+36x-8$$

Simplify the expression $(5x+1)^2-(5x+1)(5x-1)$

$$(5x+1)^2-(5x+1)(5x-1)=25x^2+10x+1-25x^2+1=10x+2$$

Simplify the expression $(2+x)^3-(2+x)(4-2x+x^2)$

$$(2+x)^3-(2+x)(4-2x+x^2)=8+12x+6x^2+x^3-8-x^3=12x+6x^2$$

2020-11-28