Video Notes - Page 10: Exponents and Binomial Expansions

Vocabulary flashcards covering exponent notation, binomial expansion, coefficients, middle term, and binomial coefficients as seen in Page 10 notes.

Exponent placeholder (■)

In the notes, ■ is used as a placeholder for an exponent in expressions like a■ or (x+2)■; for example, (x+2)■ with ■=4 expands to (x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16.

Binomial expansion

The expansion of (a+b)^n into a sum of terms with binomial coefficients: (a+b)^n = sum_{k=0}^n C(n,k) a^{n−k} b^k.

Coefficient

The numerical factor of a term in an expansion; in (a+b)^n, the coefficient of a^{n−k} b^k is C(n,k) (the binomial coefficient).

Middle term (binomial expansion)

The central term(s) in the expansion of (x+y)^n. When n is even, the middle term is T_{n/2+1}; for example, in (x+y)^6, the middle term is 20x^3y^3.

Binomial coefficient

The combinatorial number C(n,k) = n! / (k!(n−k)!) that multiplies a^{n−k} b^k in the expansion of (a+b)^n; e.g., C(6,3) = 20.

Exponent placeholder (■)

In the notes, ■ is used as a placeholder for an exponent in expressions like a■ or (x+2)■; for example, (x+2)■ with ■=4 expands to (x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16.

Binomial expansion

The expansion of (a+b)^n into a sum of terms with binomial coefficients: (a+b)^n = \sum_{k=0}^n C(n,k) a^{n−k} b^k.

Coefficient

The numerical factor of a term in an expansion; in (a+b)^n, the coefficient of $a^{n−k} b^k is $C(n,k) (the binomial coefficient).

Middle term (binomial expansion)

The central term(s) in the expansion of (x+y)^n. When n is even, the middle term is T_{n/2+1}; for example, in (x+y)^6, the middle term is 20x^3y^3.

Binomial coefficient

The combinatorial number C(n,k) = n! / (k!(n−k)!) that multiplies a^{n−k} b^k in the expansion of (a+b)^n; e.g., C(6,3) = 20.

Pascal's Triangle

A triangular array of numbers where each number is the sum of the two numbers directly above it, providing the binomial coefficients for the expansion of (a+b)^n. The n-th row contains the coefficients for (a+b)^n.

Number of terms (binomial expansion)

The total number of terms in a binomial expansion of (a+b)^n is n+1. For example, (a+b)^3 has 4 terms.

General term (binomial expansion)

The $(r+1)$-th term in the binomial expansion of (a+b)^n is given by the formula T_{r+1} = C(n,r)a^{n-r}b^r.

Exponent placeholder (■)

In the notes, ■ is used as a placeholder for an exponent in expressions like a■ or (x+2)■; for example, (x+2)■ with ■=4 expands to (x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16.

Binomial expansion

The expansion of (a+b)^n into a sum of terms with binomial coefficients: (a+b)^n = \sum_{k=0}^n C(n,k) a^{n−k} b^k.

Coefficient

The numerical factor of a term in an expansion; in (a+b)^n, the coefficient of $a^{n−k} b^k is $C(n,k) (the binomial coefficient).

Middle term (binomial expansion)

The central term(s) in the expansion of (x+y)^n. When n is even, the middle term is T_{n/2+1}; for example, in (x+y)^6, the middle term is 20x^3y^3.

Binomial coefficient

The combinatorial number C(n,k) = n! / (k!(n−k)!) that multiplies a^{n−k} b^k in the expansion of (a+b)^n; e.g., C(6,3) = 20.

Pascal's Triangle

A triangular array of numbers where each number is the sum of the two numbers directly above it, providing the binomial coefficients for the expansion of (a+b)^n. The n-th row contains the coefficients for (a+b)^n.

Number of terms (binomial expansion)

The total number of terms in a binomial expansion of (a+b)^n is n+1. For example, (a+b)^3 has 4 terms.

General term (binomial expansion)

The $(r+1)$-th term in the binomial expansion of (a+b)^n is given by the formula T_{r+1} = C(n,r)a^{n-r}b^r.

Symmetry of Binomial Coefficients

For any positive integers n and k where 0 \le k \le n, the binomial coefficients are symmetric: C(n,k) = C(n, n-k). For example, C(5,2) = C(5,3) = 10.

Sum of Exponents in Each Term (Binomial Expansion)

In the expansion of (a+b)^n, the sum of the exponents of a and b in any term is always equal to n. For example, in (x+y)^6, each term (e.g., 20x^3y^3) has exponents that sum to 6 (3+3=6).

Exponent placeholder (■)

In the notes, ■ is used as a placeholder for an exponent in expressions like a■ or (x+2)■; for example, (x+2)■ with ■=4 expands to (x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16.

Binomial expansion

The expansion of (a+b)^n into a sum of terms with binomial coefficients: (a+b)^n = \sum_{k=0}^n C(n,k) a^{n−k} b^k.

Coefficient

The numerical factor of a term in an expansion; in (a+b)^n, the coefficient of $a^{n−k} b^k is $C(n,k) (the binomial coefficient).

Middle term (binomial expansion)

The central term(s) in the expansion of (x+y)^n. When n is even, the middle term is T_{n/2+1}; for example, in (x+y)^6, the middle term is 20x^3y^3.

Binomial coefficient

The combinatorial number C(n,k) = n! / (k!(n−k)!) that multiplies a^{n−k} b^k in the expansion of (a+b)^n; e.g., C(6,3) = 20.

Pascal's Triangle

A triangular array of numbers where each number is the sum of the two numbers directly above it, providing the binomial coefficients for the expansion of (a+b)^n. The n-th row contains the coefficients for (a+b)^n.

Number of terms (binomial expansion)

The total number of terms in a binomial expansion of (a+b)^n is n+1. For example, (a+b)^3 has 4 terms.

General term (binomial expansion)

The $(r+1)$-th term in the binomial expansion of (a+b)^n is given by the formula T_{r+1} = C(n,r)a^{n-r}b^r.

Symmetry of Binomial Coefficients

For any positive integers n and k where 0 \le k \le n, the binomial coefficients are symmetric: C(n,k) = C(n, n-k). For example, C(5,2) = C(5,3) = 10.

Sum of Exponents in Each Term (Binomial Expansion)

In the expansion of (a+b)^n, the sum of the exponents of a and b in any term is always equal to n. For example, in (x+y)^6, each term (e.g., 20x^3y^3) has exponents that sum to 6 (3+3=6).

Sum of Binomial Coefficients

The sum of all binomial coefficients for a given n in the expansion of (a+b)^n is 2^n. Mathematically, \sum_{k=0}^n C(n,k) = 2^n.

Alternating Sum of Binomial Coefficients

When n > 0, the alternating sum of binomial coefficients is 0. Mathematically, \sum_{k=0}^n (-1)^k C(n,k) = 0.