Q.206 - Intermediate Algebra

Question

In the following exercises, multiply using a) the Distributive Property; b) the Vertical Method

$\left(x+5\right)\left(x^2+4x+3\right)$

Step-by-Step Solution

Answer

The product of the given polynomials is $x^3+9x^2+23x+15$

Step 1. Given Information

The given polynomials are $\left(x+5\right)\left(x^2+4x+3\right)$ 

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

Part (a) Step 1. Using distributive property

According to distributive property:

$$\begin{gathered} \left(x+5\right)\left(x^2+4x+3\right)=x\left(x^2+4x+3\right)+5\left(x^2+4x+3\right) \\ \left(x+5\right)\left(x^2+4x+3\right)=x^3+4x^2+3x+5x^2+20x+15 \\ \left(x+5\right)\left(x^2+4x+3\right)=x^3+9x^2+23x+15 \end{gathered}$$

Part (a) Step 2. Conclusion

The product of the polynomials is $x^3+9x^2+23x+15$

Part (b) Step 1. Using Vertical Method

Multiplying using vertical method:

$\begin{array}{rrr}{x}^2+& 4x+& 3\\ \times & x+& 5\\ 5x^2+& 20x+& 15\\ x^3+4x^2+& 3x & \\ x^3+9x^2+& 23x+& 15\end{array}$

Part (b) Step 2. Conclusion

The product of the polynomials is $x^3+9x^2+23x+15$