Flashcards on Binomial Theorem

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What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand the powers of a binomial expression.

State the binomial theorem.

The binomial theorem states that for any positive integer n, the expansion of (a + b)^n can be found using the formula: (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCr * a^(n-r) * b^r + ... + nCn * a^0 * b^n, where nC0, nC1, ..., nCr represents the binomial coefficients.

What are binomial coefficients?

Binomial coefficients are the coefficients obtained from the expansion of (a + b)^n using the binomial theorem. These coefficients represent the numerical coefficients of each term in the expansion.

How do you find the coefficient of a specific term in the expansion of a binomial expression?

The coefficient of a specific term in the expansion of a binomial expression can be found using the formula: coefficient = nCr * a^(n-r) * b^r, where n is the power of the binomial expression, r is the power of b in the specific term, and nCr is the binomial coefficient.

Expand (x + y)^3 using the binomial theorem.

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

What is the formula for expanding a binomial to the power of 4?

The formula for expanding a binomial to the power of 4 is: (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.

What is the general formula for expanding a binomial expression (a + b)^n?

The general formula for expanding a binomial expression (a + b)^n using the binomial theorem is: (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCr * a^(n-r) * b^r + ... + nCn * a^0 * b^n.

What are the properties of the binomial coefficients?

The properties of the binomial coefficients include symmetry, Pascal's triangle, and the summation property.

What is Pascal's triangle?

Pascal's triangle is an arrangement of numbers in the shape of a triangle, where each number is the sum of the two numbers directly above it. It is used to calculate binomial coefficients.

How many terms are there in the expansion of (a + b)^n?

The expansion of (a + b)^n will have (n + 1) terms.

Expand (2x - 3y)^2 using the binomial theorem.

(2x - 3y)^2 = 4x^2 - 12xy + 9y^2

What is the coefficient of the middle term in the expansion of (a + b)^n when n is even?

When n is even, the coefficient of the middle term in the expansion of (a + b)^n is given by nC(n/2)

What is the coefficient of the second term in the expansion of (a + b)^n?

The coefficient of the second term in the expansion of (a + b)^n is given by nC1.

What is the coefficient of the last term in the expansion of (a + b)^n?

The coefficient of the last term in the expansion of (a + b)^n is given by nCn or 1.

Expand (2a - b)^3 using the binomial theorem.

(2a - b)^3 = 8a^3 - 12a^2b + 6ab^2 - b^3

What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand the powers of a binomial expression.

State the binomial theorem.

The binomial theorem states that for any positive integer n, the expansion of (a + b)^n can be found using the formula: (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCr * a^(n-r) * b^r + ... + nCn * a^0 * b^n, where nC0, nC1, ..., nCr represents the binomial coefficients.

What are binomial coefficients?

Binomial coefficients are the coefficients obtained from the expansion of (a + b)^n using the binomial theorem. These coefficients represent the numerical coefficients of each term in the expansion.

How do you find the coefficient of a specific term in the expansion of a binomial expression?

The coefficient of a specific term in the expansion of a binomial expression can be found using the formula: coefficient = nCr * a^(n-r) * b^r, where n is the power of the binomial expression, r is the power of b in the specific term, and nCr is the binomial coefficient.

Expand (x + y)^3 using the binomial theorem.

(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3

What is the formula for expanding a binomial to the power of 4?

The formula for expanding a binomial to the power of 4 is: (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.

What is the general formula for expanding a binomial expression (a + b)^n?

The general formula for expanding a binomial expression (a + b)^n using the binomial theorem is: (a + b)^n = nC0 * a^n * b^0 + nC1 * a^(n-1) * b^1 + nC2 * a^(n-2) * b^2 + ... + nCr * a^(n-r) * b^r + ... + nCn * a^0 * b^n.

What are the properties of the binomial coefficients?

The properties of the binomial coefficients include symmetry, Pascal's triangle, and the summation property.

What is Pascal's triangle?

Pascal's triangle is an arrangement of numbers in the shape of a triangle, where each number is the sum of the two numbers directly above it. It is used to calculate binomial coefficients.

How many terms are there in the expansion of (a + b)^n?

The expansion of (a + b)^n will have (n + 1) terms.

Expand (2x - 3y)^2 using the binomial theorem.

(2x - 3y)^2 = 4x^2 - 12xy + 9y^2

What is the coefficient of the middle term in the expansion of (a + b)^n when n is even?

When n is even, the coefficient of the middle term in the expansion of (a + b)^n is given by nC(n/2)

What is the coefficient of the second term in the expansion of (a + b)^n?

The coefficient of the second term in the expansion of (a + b)^n is given by nC1.

What is the coefficient of the last term in the expansion of (a + b)^n?

The coefficient of the last term in the expansion of (a + b)^n is given by nCn or 1.

Expand (2a - b)^3 using the binomial theorem.

(2a - b)^3 = 8a^3 - 12a^2b + 6ab^2 - b^3

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