The simplified form of R(x) is [tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]
Simplifying an expression
From the question, we are to simplify the expression
From the given information,
[tex]f(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30}[/tex]
and
[tex]g(x) =\frac{x^{2}-9x+20 }{x^{2}-12x+36}[/tex]
Also,
[tex]R(x) = f(x) \div g(x)[/tex]
∴ [tex]R(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30} \div \frac{x^{2}-9x+20 }{x^{2}-12x+36}[/tex]
[tex]R(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30} \times \frac{x^{2}-12x+36 }{x^{2}-9x+20}[/tex]
Factoring each of the quadratics
[tex]R(x) =\frac{x^{2}-7x-4x+28 }{x^{2}-6x-5x+30} \times \frac{x^{2}-6x-6x+36 }{x^{2}-4x-5x+20}[/tex]
[tex]R(x) =\frac{x(x-7)-4(x-7) }{x(x-6)-5(x-6)} \times \frac{x(x-6)-6(x-6) }{x(x-4)-5(x-4)}[/tex]
[tex]R(x) =\frac{(x-4)(x-7)}{(x-5)(x-6)} \times \frac{(x-6)(x-6)}{(x-5)(x-4)}[/tex]
Simplifying
[tex]R(x) =\frac{(x-7)}{(x-5)} \times \frac{(x-6)}{(x-5)}[/tex]
[tex]R(x) =\frac{(x-7)(x-6)}{(x-5)(x-5)}[/tex]
[tex]R(x) =\frac{x^{2} -6x-7x+42}{x^{2} -5x-5x+25}[/tex]
[tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]
Hence, the simplified form of R(x) is [tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]
Learn more on Simplifying an expression here: https://brainly.com/question/1280754
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