Question

In the following exercises, multiply the binomials using a) the Distributive Property; b) the FOIL method; c) the Vertical Method

$\left(y+9\right)\left(y+3\right)$

Short answer

The product of the given binomials is$y^2+12y+27$

Step-by-step solution

Step 1. Given Information

The given binomials are $\left(y+9\right)\left(y+3\right)$

The product is to be determined using a) the Distributive Property; b) the FOIL method; c) the Vertical Method

Part (a) Step 1. Using distributive property

According to distributive property:

$$\begin{gathered} \left(y+9\right)\left(y+3\right)=y\left(y+3\right)+9\left(y+3\right) \\ \left(y+9\right)\left(y+3\right)=y^2+3y+9y+27 \\ \left(y+9\right)\left(y+3\right)=y^2+12y+27 \end{gathered}$$

Part (a) Step 2. Conclusion

The product of the binomials is$y^2+12y+27$

Part (b) Step 1. Using FOIL Method

Using FOIL method, the terms are multiplied in the order of First Terms, Outer Terms, Inner Terms and then Last Terms:

$$\begin{gathered} \left(y+9\right)\left(y+3\right)=\left(y\right)\left(y\right)+\left(y\right)\left(3\right)+\left(9\right)\left(y\right)+\left(9\right)\left(3\right) \\ \left(y+9\right)\left(y+3\right)=y^2+3y+9y+27 \\ \left(y+9\right)\left(y+3\right)=y^2+12y+27 \end{gathered}$$

Part (b) Step 2. Conclusion

The product of the binomials is$y^2+12y+27$

Part (c) Step 1. Using Vertical Method

Multiplying the binomials using vertical method:

$\begin{array}{rrr}& y& +9\\ \times & y& +3\\ & 3y& +27\\ y^2& +9y& \\ y^2& +12y& +27\end{array}$

Part (c) Step 2. Conclusion

The product of the binomials is$y^2+12y+27$