Pascal coefficients and finite binomial expansion — 0606
Use binomial coefficients to expand an integer power exactly.
Notes and key methods
For a nonnegative integer n, \((a+bx)^n=\sum_{r=0}^n {n\choose r}a^{n-r}(bx)^r\). Each term has total exponent n across the two selected components.
The finite expansion is valid for every real x. The coefficient of \(\displaystyle x^{r}\) excludes \(\displaystyle x^{r}\) itself. When multiplying two expansions, collect every product that contributes the required total power.
General terms with positive and negative powers
For integer \(n\ge0\), the term indexed by r in \((ax^p+bx^q)^n\) is \(\binom nr a^{n-r}b^r x^{p(n-r)+qr}\), with \(r=0,1,\ldots,n\). A term independent of x requires exponent zero and a permitted integer r.
In \((2x^3+3x^{-2})^5\), the exponent is \(15-5r\). Thus \(\displaystyle r=3\) gives the constant term \(\binom{5}{3}2^2 3^3=1080\). This finite identity is valid for \(\displaystyle x\ne 0\); it has no infinite-series convergence restriction.
Check the conditions
The exponent n is a nonnegative integer here.
Collect coefficients after accounting for powers of b.
Three worked examples
Example 1 · Finite binomial expansions
Consider \(\left(4 x + 3\right)^{6}\). Work without a calculator.
Expand through the term in \(\displaystyle x^{3}\).
State where the complete finite expansion is valid.
Find the coefficient of \(\displaystyle x^2\).
Find the coefficient of \(\displaystyle x^3\).
Show the reasoning
(a)
Apply the binomial theorem for \(\displaystyle r=0,1,2,3\); the remaining terms have degree at least four.
Answer: \(34560 x^{3} + 19440 x^{2} + 5832 x + 729\) + 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.
\(19440\)
Answer: \(19440\).
(d)
Use \(\displaystyle \binom{n}{r} a^{n-r}b^{r}\) for the selected power.
\(34560\)
Answer: \(34560\).
Example 2 · Finite binomial expansions
Consider \(\left(5 x + 1\right)^{7}\). Work without a calculator.
Expand through the term in \(\displaystyle x^{3}\).
State where the complete finite expansion is valid.
Find the coefficient of \(\displaystyle x^2\).
Find the coefficient of \(\displaystyle x^3\).
Show the reasoning
(a)
Apply the binomial theorem for \(\displaystyle r=0,1,2,3\); the remaining terms have degree at least four.
Answer: \(4375 x^{3} + 525 x^{2} + 35 x + 1\) + 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.
\(525\)
Answer: \(525\).
(d)
Use \(\displaystyle \binom{n}{r} a^{n-r}b^{r}\) for the selected power.
\(4375\)
Answer: \(4375\).
Example 3 · Finite binomial expansions
Consider \(\left(2 x + 2\right)^{4}\). Work without a calculator.
Expand through the term in \(\displaystyle x^{3}\).
State where the complete finite expansion is valid.
Find the coefficient of \(\displaystyle x^2\).
Find the coefficient of \(\displaystyle x^3\).
Show the reasoning
(a)
Apply the binomial theorem for \(\displaystyle r=0,1,2,3\); the remaining terms have degree at least four.
Answer: \(64 x^{3} + 96 x^{2} + 64 x + 16\) + 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.
\(96\)
Answer: \(96\).
(d)
Use \(\displaystyle \binom{n}{r} a^{n-r}b^{r}\) for the selected power.
\(64\)
Answer: \(64\).
Common mistakes
Do not omit \(\displaystyle b^{r}\) from the r-th coefficient.
A finite integer-power expansion has no |x|<1 restriction.