Q15. - Algebra 1

Question

AREA The area of a rectangle is $x^2+5x+6$. What binomial expressions represent the side lengths of the rectangle?

Step-by-Step Solution

Answer

The length and width of the rectangle are $\left(x+3\right)$ and $\left(x+2\right)$.

Step 1. Write the formula for area of rectangle.

Area of rectangle is given by $A=l\times w$.

Step 2. Find the GCF.

Find the GCF of $x^2,5x$ and 6.

$$\begin{gathered} x^2=1\cdot x\cdot x \\ 5x=1\cdot 5\cdot x \\ 6=1\cdot 2\cdot 3 \end{gathered}$$

Therefore, GCF is 1.

Step 3. Apply factoring pattern of a x 2 + b x + c .

There are two numbers having product of 6 and sum of $5$. The numbers are $2$ and $3$.

$x^2+mx+nx+6=x^2+2x+3x+6$

Step 4. Group terms with common factors and factor out GCF.

Group terms with common factors and factor out GCF from each grouping.

$$\begin{gathered} \left(x^2+2x\right)+\left(3x+6\right)=x\left(x+2\right)+3\left(x+2\right) \\ =\left(x+3\right)\left(x+2\right) \end{gathered}$$

The factors of $x^2+5x+6$ are $\left(x+3\right)\left(x+2\right)$. Therefore, length and width of the rectangle are $\left(x+3\right)$ and $\left(x+2\right)$.