Factoring out and cancelling

Every term of $3h^3+6h^2+5h$ contains $h$, so it factors as $h(3h^2+6h+5)$. Inside the parentheses is each term divided by $h$.

If every term of the numerator has an $h$, it cancels with the $h$ in the denominator. If even one term has no $h$, as in $\dfrac{h^2+2h+1}{h}$, the $h$ in the denominator does not go away.

Pull out numerical common factors too. For $10x^2+25x$, take out the greatest common divisor $5$ of $10$ and $25$ together with $x$: $5x(2x+5)$.

Examples

1. 01Factor out h. What expression goes in □?$\displaystyle 3h^3+6h^2+5h = h(\square)$AnswerHide

   Answer$\displaystyle 3h^2+6h+5$

   Steps

   1. $\displaystyle 3h^3+6h^2+5h$$\displaystyle = h \cdot 3h^2 + h \cdot 6h + h \cdot 5$
   2. Divide by h
   3. $\displaystyle \square$$\displaystyle = 3h^2+6h+5$
2. 02In which one does the h in the denominator cancel out?$\displaystyle$A$\displaystyle \frac{h^2+9}{h}$B$\displaystyle \frac{h^3+2}{h^2}$C$\displaystyle \frac{h^2+2h}{h^2}$D$\displaystyle \frac{h^2+2h}{h}$AnswerHide

   Answer$\displaystyle \frac{h^2+2h}{h}$

   Steps

   1. $\displaystyle \frac{h^2+2h}{h}$$\displaystyle = \frac{h(h+2)}{h}$$\displaystyle = h+2$
3. 03Factor out every common factor:$\displaystyle 10x^2+25x$AnswerHide

   Answer$\displaystyle 5x(2x+5)$

   Steps

   1. $\displaystyle 10x^2+25x$$\displaystyle = 5x \cdot 2x + 5x \cdot 5$$\displaystyle = 5x(2x+5)$
4. 04Factor out h. What expression goes in □?$\displaystyle 3h^3+3h^2+2h = h(\square)$AnswerHide

   Answer$\displaystyle 3h^2+3h+2$

   Steps

   1. $\displaystyle 3h^3+3h^2+2h$$\displaystyle = h \cdot 3h^2 + h \cdot 3h + h \cdot 2$
   2. Divide by h
   3. $\displaystyle \square$$\displaystyle = 3h^2+3h+2$
5. 05Factor out h. What expression goes in □?$\displaystyle 3h^3+5h^2+5h = h(\square)$AnswerHide

   Answer$\displaystyle 3h^2+5h+5$

   Steps

   1. $\displaystyle 3h^3+5h^2+5h$$\displaystyle = h \cdot 3h^2 + h \cdot 5h + h \cdot 5$
   2. Divide by h
   3. $\displaystyle \square$$\displaystyle = 3h^2+5h+5$
6. 06Factor out h. What expression goes in □?$\displaystyle 2h^3+5h^2+4h = h(\square)$AnswerHide

   Answer$\displaystyle 2h^2+5h+4$

   Steps

   1. $\displaystyle 2h^3+5h^2+4h$$\displaystyle = h \cdot 2h^2 + h \cdot 5h + h \cdot 4$
   2. Divide by h
   3. $\displaystyle \square$$\displaystyle = 2h^2+5h+4$

FAQ

Q1Is $x(10x+25)$ wrong?AnswerHide

A

It is equal, but the parentheses still contain a common factor $5$. Factoring out everything means going on to $5x(2x+5)$.

Q2Why do we need $h\neq 0$ to cancel?AnswerHide

A

Dividing by $h$ is impossible when $h=0$. In the definition of the derivative, you also cancel while $h\neq 0$ and only then let $h$ approach $0$.