Question

Factorise the expression and divide them as directed: $\left(y^2+7y+10\right)÷\left(y+5\right)$

Answer 1

Step 1: Factorize ${\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{7}\mathit{y}\mathbf{+}\mathbf{10}$

• By splitting the middle term, we need to find the two numbers such that, their

$$\begin{gathered} sum=7 \\ product=10 \end{gathered}$$

• Let us consider the two numbers $2$and $5$, then

$$\begin{gathered} Sum=2+5=7 \\ Product=2\times 5=10 \end{gathered}$$

• so, we can write $y^2+7y+10=y+2y+5y+10$

$$\begin{gathered} =y\left(y+2\right)+5\left(y+2\right) \\ \Rightarrow y^2+7y+10=\left(y+2\right)\left(y+5\right) \end{gathered}$$

Step 2: Divide ${\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{7}\mathit{y}\mathbf{+}\mathbf{10}$ by $\mathit{y}\mathbf{+}\mathbf{5}$

$$\begin{gathered} \frac{y^2+7y+10}{y+5}=\frac{\left(y+2\right)\left(y+5\right)}{y+5} \\ =y+2 \end{gathered}$$

Hence, $\mathbf{(}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{+}\mathbf{10}\mathbf{)}\mathbf{÷}\mathbf{(}\mathbf{y}\mathbf{+}\mathbf{5}\mathbf{)}\mathbf{=}\mathit{y}\mathbf{+}\mathbf{2}$