Simplify
𝑓
(
𝑥
)
=
𝑥
2
+
13
𝑥
+
36
𝑥
2
+
8
𝑥
−
9
, under the assumption that
𝑥
≠
−
9.
(Hint: factorise first!)

Question

Simplify 
𝑓
(
𝑥
)
=
𝑥
2
+
13
𝑥
+
36
𝑥
2
+
8
𝑥
−
9
, under the assumption that 
𝑥
≠
−
9.
 (Hint: factorise first!)

BlackTom AI · Accepted Answer

Factor the quadratics: x^2 + 13x + 36 = (x + 4)(x + 9), and x^2 + 8x − 9 = (x + 9)(x − 1).

Step-by-step:
1. Write the original expression as a fraction of factors:
   f(x) = (x^2 + 13x + 36) / (x^2 + 8x − 9)
        = (x + 4)(x + 9) / [(x + 9)(x − 1)].
2. Cancel the common nonzero factor (x + 9):
   f(x) = (x + 4) / (x − 1), provided (x + 9) ≠ 0 when canceling.
3. Note domain restrictions from the original denominator: x^2 + 8x − 9 = (x + 9)(x − 1) so the original expression is undefined at x = −9 and x = 1. Even though cancellation produces (x + 4)/(x − 1), the original function is not defined at x = −9 (and also not at x = 1). The condition x ≠ −9 given in the problem ensures we do not divide by zero when canceling; one should also remember x ≠ 1 to respect the original domain.

Why this is correct:
- Factorization shows the numerator and denominator share exactly the factor (x + 9), so cancellation reduces the rational expression to (x + 4)/(x − 1).
- This reduced form produces the same values as the original for all x except where the original denominator is zero (x = −9, 1).

Why other choices would be incorrect:
- Any alternative simplified form that does not come from correct factoring (for example, failing to factor or cancelling the wrong factor) will not match the algebraic cancellation above.
- Expressions that ignore the (x − 1) in the denominator are missing a necessary restriction or term.

Conclusion: The simplified form is f(x) = (x + 4)/(x − 1), which matches the provided answer (d) f(x) = x+4 over x−1 (with the understanding that x ≠ −9 and x ≠ 1).

Simplify 𝑓 ( 𝑥 ) = 𝑥 2 + 13 𝑥 + 36 𝑥 2 + 8 𝑥 − 9 , under the assumption that 𝑥 ≠ − 9. (Hint: factorise first!)单项选择题

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