The value of q(x) is [tex]2 x+5[/tex]
The value of r(x) is [tex]6[/tex]
Explanation:
The given expression is [tex]\frac{2 x^{2}+13 x+26}{x+4}[/tex]
We need to rewrite the expression in the form of [tex]q(x)+\frac{r(x)}{b(x)}[/tex]
Simplifying the expression, we get,
[tex]\frac{2 x^{2}+8 x+5x+26}{x+4}[/tex]
Separating the fractions, we have,
[tex]\frac{2 x^{2}+8 x}{x+4}+\frac{5 x+26}{x+4}[/tex]
[tex]2 x+\frac{5 x+26}{x+4}[/tex] -----------(1)
Now, we shall further simplify the term [tex]\frac{5 x+26}{x+4}[/tex] , we get,
[tex]\frac{5 x+26}{x+4}=\frac{5 x+20}{x+4}+\frac{6}{x+4}[/tex]
Common out 5 from the numerator, we have,
[tex]\frac{5 x+26}{x+4}=5+\frac{6}{x+4}[/tex]
Substituting the value [tex]\frac{5 x+26}{x+4}=5+\frac{6}{x+4}[/tex] in the equation(1), we get,
[tex]2 x+5+\frac{6}{x+1}[/tex]
Thus, the expression [tex]\frac{2 x^{2}+13 x+26}{x+4}=2 x+5+\frac{6}{x+1}[/tex] is in the form of [tex]q(x)+\frac{r(x)}{b(x)}[/tex]
Hence, we have,
[tex]q(x)=2 x+5[/tex]
[tex]r(x)=6[/tex] and
[tex]b(x)=x+4[/tex]
