Question

In the following exercises, multiply the binomials using

(a) the Distributive Property;

(b) the FOIL method;

(c) the Vertical Method.

$\left(w+5\right)\left(w+7\right)$

Short answer

The solution is

$\left(w+5\right)\left(w+7\right)=w^2+12w+35$

Step-by-step solution

Step 1. Given expression

The given expression is

$\left(w+5\right)\left(w+7\right)$

Part (a) Step 1. Apply the concept of Distributive property

are any real numbers or algebraic expressions, then

$\left(A+B\right)\left(C+D\right)=A\left(C+D\right)+B\left(C+D\right)=AC+AD+BC+BD$

Part (a) Step 2. Simplify using Distributive property

Using Distributive property,$$\begin{gathered} \left(w+5\right)\left(w+7\right) \\ =w\left(w+7\right)+5\left(w+7\right) \\ =w^2+7w+5w+35 \\ =w^2+12w+35 \end{gathered}$$

Part (b) Step 1. Apply the concept of Foil method

If $A,B,CandD$are any algebraic expressions then, $\left(A+B\right)\left(C+D\right)$can be simplified by Foil property using following steps

• Multiply the First terms $A$and$C$
• Multiply the Outer Terms $AandD$
• Multiply the last terms $BandD$
• Combine like terms, when possible

Part (b) Step 2. Simplify using Foil method

Using Foil method,

$\left(w+5\right)\left(w+7\right)$

• $\left(w+5\right)\left(w+7\right)=w^2+\_+\_+\_$
  
• $\left(w+5\right)\left(w+7\right)=w^2+7w+\_+\_$
  
• $\left(w+5\right)\left(w+7\right)=w^2+7w+5w+\_$
  
• $\left(w+5\right)\left(w+7\right)=w^2+7w+5w+35$
  
• $\left(w+5\right)\left(w+7\right)=w^2+12w+35$
  

Part (c) Step 1. Apply the concept of Vertical method 

To multiply two polynomials we use vertical method

Part (c) Step 2. Simplify using vertical method 

Using vertical method,

$$\begin{gathered} w+5 \\ \frac{\times w+7}{7w+35}partialproduct \\ \frac{w^2+5w}{w^2+12w+35}partialproduct \end{gathered}$$