Question

How to write for rational function with x intercept at x=6, and x=5?

Answer 1

Hint: Here in the given question we need to build the equation whose factors or intercepts the given values are present in the question, for which we have to construct the algebraic equation, by writing the variable In the form that on equating to zero they give the same factor which are present in the question.

Formulae Used:
\[f(x) = 0\]for finding the intercept for any equation of variable “x” with the function of “f(x)”.

Complete step-by-step solution:
Here in the given question we have to find the equation, for the given intercept as x=5,6
Here we have to know that an intercept occurs when \[f(x) = 0\]. Here we are just looking for a rational functional \[f(x)\], such that the given condition follows by the equation, on solving we get:
\[ \Rightarrow f(6) = f(5) = 0\]
This condition need to be followed by the constructed equation, now here we can construct the equation as:
\[ \Rightarrow f(x) = (x - 6)(x - 5)\]
Here we can see that the given equation gives the factor as “5,6” on equating to zero, hence we have our required equation just by expanding the brackets, on solving we get:
\[ \Rightarrow f(x) = x(x - 5) - 6(x - 5) = {x^2} - 5x - 6x + 30 = {x^2} - 11x + 30\]

Hence we got the final desired equation whose intercept for “x” are “5 and 6”, hence it is our final required equation.

Note: Here for the given question we can do factorization method which involves that the given intercepts are roots of the equation and then by assuming the variable we can form the equation, so to factorize them and get the roots of the equation.