Binomial Expansion — Maths4U Original

Notes

Before you start: multiply brackets, use integer powers and collect like terms.

For a nonnegative integer n, the binomial theorem is the exact finite identity

\[(a+b)^n=\sum_{r=0}^n\binom nr a^{n-r}b^r,\qquad \binom nr=\frac{n!}{r!(n-r)!}.\]

Here 0!=1 and the index r runs through integers 0,…,n. For n=0 the expansion is the constant 1 as a polynomial identity. This block uses finite expansions only: it does not apply an infinite-series rule to a negative or fractional exponent.

In \((a+bx)^n\), choose the second term from r of the n factors. There are \(\binom nr\) such choices. The powers of a decrease as the powers of bx increase. In \((a-bx)^n\), odd r gives a minus sign and even r a plus sign; the Pascal coefficient itself remains nonnegative.

“First three terms in ascending powers of x” means the constant, x and x² terms, when present. Keep the power of the constant a: the x² coefficient is \(\binom n2 a^{n-2}b^2\), not just the Pascal number.

Recall later

\((1-2x)^3=1-6x+12x^2-8x^3\). The coefficients of the even powers are positive, while those of the odd powers are negative.

Common mistakes

• Dropping the power of the constant part.
• Alternating the signs of Pascal numbers instead of applying the power to the entire negative term.
• Using a finite integer-power formula for a negative or fractional exponent.

Notes

Before you start: use the binomial theorem and the laws of indices, including negative powers.

For \((a+bx)^n\), the general term in binomial order is \(T_{r+1}=\binom nr a^{n-r}b^r x^r\), for integers 0≤r≤n. The first term uses r=0, so the fourth term uses r=3. A coefficient is the factor multiplying a power of x; a term includes that power.

For integer p and q in \((Ax^p+Bx^q)^n\), combine the powers before solving for an index:

\[T_{r+1}=\binom nr A^{n-r}B^r x^{p(n-r)+qr}.\]

Set the exponent equal to the requested power. A solution contributes only when r is an integer in [0,n]. If no such r exists, that coefficient is zero. For negative powers, x≠0; a “constant term” has exponent zero and is not found by substituting x=0 into an undefined expression.

Recall later

The coefficient of x² in \((1+2x)^5\) is 40; its third term is 40x². These are different answers to different requests.

Common mistakes

• Confusing the index r with the term number r+1.
• Accepting a fractional or out-of-range index as a contributing term.
• Substituting zero to find a constant term when negative powers make the expression undefined there.

Notes

Before you start: find a binomial coefficient and multiply polynomials.

If \(A(x)=\sum a_i x^i\) and \(B(x)=\sum b_j x^j\) are polynomials in nonnegative powers, the coefficient of xᵐ in A(x)B(x) is \(\sum_{i=0}^m a_i b_{m-i}\). List all power pairs that add to m. For m=2 these are (0,2), (1,1) and (2,0); the middle product is easy to miss.

For ordinary polynomials, powers higher than m cannot contribute to xᵐ, so it is safe to truncate each factor above that degree. This shortcut needs care with negative powers: a high positive power can combine with a negative one to give the target.

Before multiplying, look for \((1+x)^n(1-x)^n=(1-x^2)^n\). This contains only even powers. More generally, if F is a polynomial, its even- and odd-power coefficient sums are \((F(1)+F(-1))/2\) and \((F(1)-F(-1))/2\). This follows because (−1) to an even power is 1 and to an odd power is −1.

Recall later

The coefficient of x² in \((1+2x)(1+3x)\) is 6, from the pair (1,1). Adding the separate x² coefficients would give the wrong answer.

Common mistakes

• Missing the cross-product when collecting a coefficient.
• Keeping only same-degree terms instead of pairs whose powers add to the target.
• Applying polynomial truncation to factors with negative powers without checking other possible pairs.

Notes

Before you start: solve linear and quadratic equations, and use the first coefficients of a binomial expansion.

For \((1+ax)^n\), the x and x² coefficients are \(c_1=na\) and \(c_2=n(n-1)a^2/2\). With n≥2 and a≠0,

\[\frac{c_2}{c_1^2}=\frac{n-1}{2n}.\]

The ratio can recover n before a. State why any division is valid. The exponent must still be a nonnegative integer in this finite-expansion setting; do not accept an algebraic value that violates the question.

An even-power coefficient may determine a² without choosing a sign. If no additional information selects a branch, both values must be retained. An odd-power coefficient or a stated sign condition can distinguish them.

Recall later

From k²=4 alone, k=−2 or k=2. Reporting only the positive root loses an admissible parameter.

Common mistakes

• Discarding the negative parameter root without a reason.
• Treating n as an unrestricted real number in a finite binomial formula.
• Dividing by a coefficient or parameter without checking that it is nonzero.

Notes

Before you start: expand integer powers, substitute decimals accurately and use units for modelled quantities.

A finite binomial expansion is exact. An approximation is introduced only when terms are omitted. Write \(F(x)=P(x)+R(x)\), where P is the retained polynomial and R is the exact omitted remainder. If R>0, P underestimates F; if R<0, it overestimates F. The absolute error is |R|.

For a positive integer n, the first terms of \((1+x)^n\) are \(1+nx+\binom n2x^2+\cdots\). A small |x| often makes later terms less important, but it does not on its own prove a requested error bound. Check the actual remainder and the stated substitution. A finite expansion has no |x|<1 convergence requirement.

For a relative change t in the side of a cube, the volume factor is \((1+t)^3\). First-order volume growth is approximately 3t, while 3t²+t³ is the discarded factor. Keep a percentage such as 2% as t=0.02, and distinguish an error in volume units from a percentage error.

Recall later

Using 1+3x+3x² for \((1+x)^3\) leaves the exact remainder x³. Its sign changes when x changes sign.

Common mistakes

• Claiming the full finite expansion is only an approximation.
• Assuming the sign of an omitted sum from its first term without checking the rest.
• Using 2 instead of 0.02 for a 2% change or mixing relative and absolute error.