Question

Part b) Rewrite each expression as an equivalent rational expression using the LCD:

Answer 1

The equivalent expression of 4 / (p² + 7p + 12) is 4 / (p + 3)(p + 4).

The equivalent expression of 3p / (p² + 8p + 15) is 3p/ (p + 3)(p + 5).

How to rewrite the expression?

The given expression is rewritten as an equivalent rational expression using the LCD as follows;

The given expression;

4 / (p² + 7p + 12)

The denominator is factorized as follows;

= p² + 7p + 12

= p² + 4p + 3p + 12

= p(p + 4) + 3(p + 4)

= (p + 3)(p + 4)

The LCD = (p + 3)(p + 4)

The equivalent expression = 4 / (p + 3)(p + 4)

The given expression;

3p / (p² + 8p + 15)

The factorize;

= p² + 8p + 15

= p² + 3p + 5p + 15

= p(p + 3) + 5(p + 3)

= (p + 3)(p + 5)

The equivalent expression = 3p/ (p + 3)(p + 5)

Answer 2

The Solution:

Given these expressions:

`\begin{gathered} (4)/(p^2+7p+12) \\ \\ \\ (3p)/(p^2+8p+15) \end{gathered}`

We are asked to use the Lowest Common Denominator to rewrite each expression as an equivalent rational expression.

Step 1:

Use the Factor Method of solving a quadratic expression to resolve each of the denominators.

`(4)/(p^(2)+7p+12)=(4)/(p^2+3p+4p+12)=(4)/(p(p+3)+4(p+3))=(4)/((p+3)(p+4))`

Step 2:

Similarly,

`(3p)/(p^(2)+8p+15)=(3p)/(p^2+3p+5p+15)=(3p)/(p(p+3)+5(p+3))=(3p)/((p+3)(p+5))`

Therefore, the correct answers are respectively:

`\begin{gathered} (4)/(p^2+7p+12)=(4)/((p+3)(p+4)) \\ \\ (3p)/(p^(2)+8p+15)=(3p)/((p+3)(p+5)) \end{gathered}`