Answer:
The simplest form of [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex] is [tex]\frac{2}{(x+1)(x+3)}[/tex]
Step-by-step explanation:
To add two algebraic fractions do these steps
To simplify [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex]
Factorize each denominator
∵ The denominator of the first fraction is x² + 5x + 6
∵ x² = (x)(x)
∵ 6 = (2)(3)
∵ (2)(x) + (3)(x) = 5x ⇒ middle term
∴ x² + 5x + 6 = (x + 2)(x + 3)
∵ The denominator of the second fraction is x² + 3x + 2
∵ x² = (x)(x)
∵ 2 = (2)(1)
∵ (2)(x) + (1)(x) = 3x ⇒ middle term
∴ x² + 3x + 2 = (x + 2)(x + 1)
Find the LCM of the two denominators
∵ The denominators are (x + 2)(x + 3) and (x + 2)(x + 1)
- LCM is all the different factors multiplied together
∴ LCM of them is (x + 1)(x + 2)(x + 3)
- Divide LCM by each denominator
∵ (x + 1)(x + 2)(x + 3) ÷ (x + 2)(x + 3) = (x + 1)
- Multiply the numerator of the first fraction by (x + 1)
∵ (x + 1)(x + 2)(x + 3) ÷ (x + 2)(x + 1) = (x + 3)
- Multiply the numerator of the second fraction by (x + 3)
∴ [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex] = [tex]\frac{x+1}{(x+1)(x+2)(x+3)}+\frac{x+3}{(x+1)(x+2)(x+3)}[/tex]
- Add the numerators and write the answer as a single fraction
∴ [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex] = [tex]\frac{2x+4}{(x+1)(x+2)(x+3)}[/tex]
- Factorize the numerator by taking 2 as a common factor
∵ 2x + 4 = 2(x + 2)
∴ [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex] = [tex]\frac{2(x+2)}{(x+1)(x+2)(x+3)}[/tex]
- Simplify the fraction by dividing up and down by (x + 2)
∴ [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex] = [tex]\frac{2}{(x+1)(x+3)}[/tex]
The simplest form of [tex]\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}[/tex] is [tex]\frac{2}{(x+1)(x+3)}[/tex]
