Question

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

$\left(7q+4\right)\left(3q-8\right)$

Short answer

The product of the given binomials is$21q^2-44q-32$

Step-by-step solution

Step 1. Given Information

The given binomials are $\left(7q+4\right)\left(3q-8\right)$

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:

$$\begin{gathered} \left(7q+4\right)\left(3q-8\right)=7q\left(3q-8\right)+4\left(3q-8\right) \\ \left(7q+4\right)\left(3q-8\right)=7q\left(3q\right)-7q\left(8\right)+4\left(3q\right)-4\left(8\right) \\ \left(7q+4\right)\left(3q-8\right)=21q^2-56q+12q-32 \\ \left(7q+4\right)\left(3q-8\right)=21q^2-44q-32 \end{gathered}$$

Part (a) Step 2. Conclusion

The product of the binomials is$21q^2-44q-32$

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:

$$\begin{gathered} \left(7q+4\right)\left(3q-8\right)=\left(7q\right)\left(3q\right)+\left(7q\right)\left(-8\right)+\left(4\right)\left(3q\right)+\left(4\right)\left(-8\right) \\ \left(7q+4\right)\left(3q-8\right)=21q^2-56q+12q-32 \\ \left(7q+4\right)\left(3q-8\right)=21q^2-44q-32 \end{gathered}$$

Part (b) Step 2. Conclusion

The product of the binomials is$21q^2-44q-32$

Part (c) Step 1. Using Vertical Method

Multiplying the binomials using vertical method:

$\begin{array}{ccc}& 7q& +4\\ \times & 3q& -8\\ & -56q& -32\\ 21q^2& +12q& \\ 21q^2& -44q& -32\end{array}$

Part (c) Step 2. Conclusion

The product of the binomials is$21q^2-44q-32$