Part 1Multiply. StartFraction 7 x minus 21 Over 3 x plus 9 EndFraction7x−213x+9 times •StartFraction 4 x plus 12 Over 6 x minus 18 EndFraction4x+126x−18 Part 1StartFraction 7 x minus 21 Over 3 x plus 9 EndFraction7x−213x+9 times •StartFraction 4 x plus 12 Over 6 x minus 18 EndFraction4x+126x−18 = [input]enter your response here (Type an integer or a fraction. Simplify your answer.)

Question

Part 1Multiply. StartFraction 7 x minus 21 Over 3 x plus 9 EndFraction7x−213x+9 times •StartFraction 4 x plus 12 Over 6 x minus 18 EndFraction4x+126x−18 Part 1StartFraction 7 x minus 21 Over 3 x plus 9 EndFraction7x−213x+9 times •StartFraction 4 x plus 12 Over 6 x minus 18 EndFraction4x+126x−18 ​= [input]enter your response here ​(Type an integer or a fraction. Simplify your​ answer.)

BlackTom AI · Accepted Answer

Factor each polynomial to find common factors: 7x − 21 = 7(x − 3), 3x + 9 = 3(x + 3), 4x + 12 = 4(x + 3), 6x − 18 = 6(x − 3).

1) Relevant concepts
- Factor polynomials to reveal common multiplicative factors.
- Multiply rational expressions by multiplying numerators and denominators.
- Cancel only nonzero common factors (respecting domain restrictions where denominators are zero).

2) Step-by-step reasoning and intermediate steps
Start with the expression:
(7x − 21)/(3x + 9) * (4x + 12)/(6x − 18)

Factor each part:
= [7(x − 3)] / [3(x + 3)] * [4(x + 3)] / [6(x − 3)]

Combine multiplication:
= [7(x − 3) * 4(x + 3)] / [3(x + 3) * 6(x − 3)]

Cancel common factors (x − 3) and (x + 3), which are valid to cancel provided they are not zero:
= (7 * 4) / (3 * 6)

Compute numerator and denominator:
7 * 4 = 28
3 * 6 = 18

Simplify the fraction 28/18 by dividing numerator and denominator by 2:
28 ÷ 2 = 14
18 ÷ 2 = 9

So the simplified value is 14/9.

3) Domain note
Original expression is undefined when any denominator equals zero: 3x + 9 = 0 → x = −3 and 6x − 18 = 0 → x = 3. Thus the simplification 14/9 holds for all x except x = −3 and x = 3.

4) Why the correct answer is correct
After factoring and canceling valid nonzero factors, only constants remain, giving 28/18 = 14/9. All algebraic steps are valid for allowed x, so 14/9 is the correct simplified value.

5) Why other answers would be incorrect
- Leaving 28/18 is not fully simplified; it should reduce to 14/9.
- Incorrect cancellation (e.g., cancelling an x term across sums or cancelling 7 with 6 directly) violates factor cancellation rules.
- Forgetting domain restrictions might lead to the false claim that the simplification holds at x = ±3.

6) Conclusion
The fully simplified result is 14/9, which matches the provided answer.

类似问题

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