Hint
The exponent n is a nonnegative integer here. Use \(\displaystyle \binom{n}{r} a^{n-r}b^{r}\) for the selected power.
Worked solution
(a)
Apply the binomial theorem for \(\displaystyle r=0,1,2,3\); the remaining terms have degree at least four.
Answer: \(354375 x^{3} + 127575 x^{2} + 25515 x + 2187\) + higher powers of x
(b)
A nonnegative integer power is a polynomial identity, so it holds for all real x.
Answer: \(\mathbb{R}\)
(c)
Use \(\displaystyle \binom{n}{r} a^{n-r}b^{r}\) for the selected power.
\(127575\)
Answer: \(127575\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Use finite binomial coefficients and the appropriate power of the inner coefficient.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not omit \(\displaystyle b^{r}\) from the r-th coefficient.
(b) · 2 marks
- 1 mark: Explain why a finite polynomial identity has no infinite-series convergence restriction.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A finite integer-power expansion has no |x|<1 restriction.
(c) · 2 marks
- 1 mark: Use \(\displaystyle \binom{n}{r} a^{n-r}b^{r}\) for the selected power.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not omit \(\displaystyle b^{r}\) from the r-th coefficient.
Total: 6 marks.