Expanding (a+b)² and (a+b)³

Don't drop the middle term $2ab$ in $(a+b)^2=a^2+2ab+b^2$. For example, $(x+5)^2=x^2+10x+25$.

The coefficients of the cube are $1,\ 3,\ 3,\ 1$. When $b$ is negative, the signs alternate $+,\ -,\ +,\ -$.

With $f(x)=x^2$, $f(x+h)=(x+h)^2=x^2+2xh+h^2$. Then $\frac{f(x+h)-f(x)}{h}=2x+h$, and letting $h\to 0$ gives $f'(x)=2x$.

Examples

1. 01Expand:$\displaystyle (x+3)^3$AnswerHide

   Answer$\displaystyle x^3+9x^2+27x+27$

   Steps

   1. $\displaystyle (x+3)^3$$\displaystyle = (x+3)^2(x+3)$$\displaystyle = (x^2+6x+9)(x+3)$
   2. Multiply term by term, then collect like terms
   3. $\displaystyle = x^3+3x^2+6x^2+18x+9x+27$$\displaystyle = x^3+9x^2+27x+27$
2. 02Which expression expands to this?$\displaystyle x^2+4x+4$AnswerHide

   Answer$\displaystyle (x+2)^2$

   Steps

   1. $\displaystyle x^2 + 2 \cdot 2 x + 2^2$$\displaystyle = (x+2)^2$
3. 03With $f(x)=x^2$, what goes in □?$\displaystyle f(x+h) = x^2 + \square h + h^2$AnswerHide

   Answer$\displaystyle 2x$
4. 04After expanding, what number goes in □?$\displaystyle (x+5)^2 = x^2 + \square x+25$AnswerHide

   Answer$\displaystyle 10$
5. 05Expand:$\displaystyle (x-2)^2$AnswerHide

   Answer$\displaystyle x^2-4x+4$

   Steps

   1. $\displaystyle (x-2)^2$$\displaystyle = (x-2)(x-2)$$\displaystyle = x^2-2x-2x+4$$\displaystyle = x^2-4x+4$
6. 06Expand:$\displaystyle (x-1)^3$AnswerHide

   Answer$\displaystyle x^3-3x^2+3x-1$

   Steps

   1. $\displaystyle (x-1)^3$$\displaystyle = (x-1)^2(x-1)$$\displaystyle = (x^2-2x+1)(x-1)$
   2. Multiply term by term, then collect like terms
   3. $\displaystyle = x^3-x^2-2x^2+2x+x-1$$\displaystyle = x^3-3x^2+3x-1$

FAQ