In this activity, you’ll use the inspection method to rewrite a rational expression, a(x)/b(x), in the form q(x) + r(x)/b(x).Answer these questions to step through the process of rewriting x^2-5x+7/x-9Part ACan the polynomial in the numerator of the expression x^2-5x+7/x-9 be factored to derive (x-9) as a factor?Answer is noPart DWhat number must be added to the numerator to get the new constant term you identified in Part C?Part EAdd the number you calculated in part D to the numerator, and then subtract the number to keep the value of the expression unchanged.Part F Rewrite the numerator so it contains a trinomial that can be faced with x-9 as a common factor, and then write it in the factored formPart GRewrite the expression you found in part F as the sum of two rational expressions with (x-9) as their common denominator Part HReduce the first fraction and write the expression in this format:A(x)/b(x) = q(x)+ r(x)/b(x)

Question

contestada

Respuesta :

ValinH163936 ValinH163936 · Answer 1 · 2022-11-03T11:11:07+01:00

The expression is:

[tex]\frac{x^2-5x+7}{x-9}[/tex]

Part B

To get -9 to -5, we need to add 4. This is important because the factored form will be something like this:

[tex]x^2-5x+7=(x-9)(x+a)[/tex]

And when we distribute it, the middle term will be the sum of -9 and a, so we if we want it to be -5 (as the given expression) a has to be 4.

Part C

Now, looking to the constant part, it will be the multiplication of -9 and a, since we know that a is 4, the constant term is:

[tex]-9\cdot4=-36[/tex]

So, we need a constant term of -36 in the numerator.

Part D

Since we already got 7 in the numerator, we have to add -43 to get it to -36.

Part E

[tex]\frac{x^2-5x+7}{x-9}=\frac{x^2-5x+7+(-43)-(-43)}{x-9}=\frac{x^2-5x-36+43}{x-9}[/tex]

Part F

[tex]\frac{x^2-5x+-36+43}{x-9}=\frac{(x-9)(x+4)+43}{x-9}[/tex]

Part G

[tex]\frac{(x-9)(x+4)+43}{x-9}=\frac{(x-9)(x+4)}{x-9}+\frac{43}{x-9}[/tex]

Part H

[tex]\frac{(x-9)(x+4)}{x-9}+\frac{43}{x-9}=x+4+\frac{43}{x-9}[/tex]

So:

[tex]\frac{x^2-5x+7}{x-9}=x+4+\frac{43}{x-9}[/tex]