Maths-

General

Easy

Question

# How are Pascal’s triangle and binomial expansion such as (a + b)^{5} related?

Hint:

### Pascal's Triangle gives us the coefficients for an expanded binomial of the form (a + b)n , where n is the row of the triangle. The Binomial Theorem tells us we can use these coefficients to find the entire expanded binomial, with a couple extra tricks thrown in.

## The correct answer is: (x + y)n

Step 1 of 1:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)^{n} , where n can be any positive integer and x, y are real numbers. Pascal Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement.

The binomial expansion of (a + b)^{5} is:

Here, the coefficients of the expansions are the elements of the sixth row of the Pascal’s triangle.

Here, the coefficients of the expansions are the elements of the sixth row of the Pascal’s triangle.

You can find the expansion of (x + y)n using both Pascal’s triangle and binomial expansion.

### Related Questions to study

Maths-

### Use binomial theorem to expand .

ANSWER:

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is ,. The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is ,. The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

### Use binomial theorem to expand .

Maths-General

ANSWER:

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is ,. The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is ,. The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Maths-

### Use polynomial identities to factor the polynomials or simplify the expressions :

ANSWER:

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

.

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Hence, the factor form is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

.

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Hence, the factor form is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

### Use polynomial identities to factor the polynomials or simplify the expressions :

Maths-General

ANSWER:

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

.

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Hence, the factor form is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

.

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Hence, the factor form is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

Maths-

### Explain why the middle term is 10x.

Step 1 of 1:

The given expression is: (x + 5)

^{2}

Here, a=x and b=5.

Here, the middle term is 10x because while expanding you have the form,

### Explain why the middle term is 10x.

Maths-General

Step 1 of 1:

The given expression is: (x + 5)

^{2}

Here, a=x and b=5.

Here, the middle term is 10x because while expanding you have the form,

Maths-

### Use binomial theorem to expand .

ANSWER:

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is , here . The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression ,we would have n+1 terms. This is something you need to keep in mind.

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is , here . The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression ,we would have n+1 terms. This is something you need to keep in mind.

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

### Use binomial theorem to expand .

Maths-General

ANSWER:

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is , here . The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression ,we would have n+1 terms. This is something you need to keep in mind.

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

The given expression is , here . The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of in the binomial equation to get the expansion:

Thus, the expansion is:

Note:

For the expansion of an expression ,we would have n+1 terms. This is something you need to keep in mind.

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Maths-

### Use polynomial identities to factor the polynomials or simplify the expressions :

ANSWER:

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to simplify the given expressions.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is: .

Note:

Polynomial identities are used to simplify or to find the prduct of expressions. It reduces space and time during solving.

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to simplify the given expressions.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is: .

Note:

Polynomial identities are used to simplify or to find the prduct of expressions. It reduces space and time during solving.

### Use polynomial identities to factor the polynomials or simplify the expressions :

Maths-General

ANSWER:

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to simplify the given expressions.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is: .

Note:

Polynomial identities are used to simplify or to find the prduct of expressions. It reduces space and time during solving.

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to simplify the given expressions.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is: .

Note:

Polynomial identities are used to simplify or to find the prduct of expressions. It reduces space and time during solving.

Maths-

### How can you use polynomial identities to rewrite expressions efficiently ?

Step 1 of 1:

A polynomial expression can be in the expanded form majority of the times. We can use the polynomial identities to factorize them in to the standard form. This will reduce time and space and enhance the quality of writing. Moreover, the polynomial identities are not just limited form variables but for numbers as well. We can use these identities to multiply higher numbers and reduce the calculation part to an extent.

### How can you use polynomial identities to rewrite expressions efficiently ?

Maths-General

Step 1 of 1:

A polynomial expression can be in the expanded form majority of the times. We can use the polynomial identities to factorize them in to the standard form. This will reduce time and space and enhance the quality of writing. Moreover, the polynomial identities are not just limited form variables but for numbers as well. We can use these identities to multiply higher numbers and reduce the calculation part to an extent.

Maths-

### Use binomial theorem to expand (2c + d)^{6}

Step 1 of 2:

The given expression is (2c + d)

^{6}, here x = 2c & y = d . The value of n=6, hence we would have 6+1=7 terms in the expression.

Step 2 of 2:

Substitute the values of (2c + d)

^{6}in the binomial expression

Thus, the expansion is: .

### Use binomial theorem to expand (2c + d)^{6}

Maths-General

Step 1 of 2:

The given expression is (2c + d)

^{6}, here x = 2c & y = d . The value of n=6, hence we would have 6+1=7 terms in the expression.

Step 2 of 2:

Substitute the values of (2c + d)

^{6}in the binomial expression

Thus, the expansion is: .

Maths-

### Use binomial theorem to expand .

ANSWER:

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

Here, the value of n=4. So, the combination permutation we would use is .

We would have 4+1=5 terms in the expansion of the expression .

Step 2 of 2:

Substitute the values of in the binomial expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

Here, the value of n=4. So, the combination permutation we would use is .

We would have 4+1=5 terms in the expansion of the expression .

Step 2 of 2:

Substitute the values of in the binomial expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

### Use binomial theorem to expand .

Maths-General

ANSWER:

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

Here, the value of n=4. So, the combination permutation we would use is .

We would have 4+1=5 terms in the expansion of the expression .

Step 2 of 2:

Substitute the values of in the binomial expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Hint:

The binomial expansion is , here .

We are asked to use binomial theorem to expand .

Step 1 of 2:

Here, the value of n=4. So, the combination permutation we would use is .

We would have 4+1=5 terms in the expansion of the expression .

Step 2 of 2:

Substitute the values of in the binomial expansion:

Thus, the expansion is:

Note:

For the expansion of an expression , we would have n+1 terms. This is something you need to keep in mind.

Maths-

### Use binomial theorem to expand (x - 1)^{7}

Step 1 of 2:

The given expression is (x - 1)

^{7}, here x = x & y = -1 . The value of n=7, hence there are 7+1=8 terms in the expressions.

Step 2 of 2:

Substitute the values of (x - 1)

^{7}in the binomial equation to get the expansion:

Thus, the expansion is:

### Use binomial theorem to expand (x - 1)^{7}

Maths-General

Step 1 of 2:

The given expression is (x - 1)

^{7}, here x = x & y = -1 . The value of n=7, hence there are 7+1=8 terms in the expressions.

Step 2 of 2:

Substitute the values of (x - 1)

^{7}in the binomial equation to get the expansion:

Thus, the expansion is:

Maths-

### Use polynomial identities to factor the polynomials or simplify the expressions :

ANSWER:

Hint:

, here a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize the given expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use polynomial identities to simplify the given expression;

The factor of the expression is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

Hint:

, here a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize the given expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use polynomial identities to simplify the given expression;

The factor of the expression is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

### Use polynomial identities to factor the polynomials or simplify the expressions :

Maths-General

ANSWER:

Hint:

, here a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize the given expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use polynomial identities to simplify the given expression;

The factor of the expression is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

Hint:

, here a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize the given expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use polynomial identities to simplify the given expression;

The factor of the expression is

Note:

The multiplication of algebraic expressions is a method of multiplying two given expressions consisting of variables and constants.

Maths-

### Use Pascal triangle to expand

ANSWER:

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

From the triangle, the seventh line would give the coefficients of the expansion of the polynomial . Thus, we have:

Note:

The expansion of the polynomial can be found using the values of

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

^{n}, where n can be any positive integer and x,y are real numbers. Pascal Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement.We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

From the triangle, the seventh line would give the coefficients of the expansion of the polynomial . Thus, we have:

Note:

The expansion of the polynomial can be found using the values of

### Use Pascal triangle to expand

Maths-General

ANSWER:

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

From the triangle, the seventh line would give the coefficients of the expansion of the polynomial . Thus, we have:

Note:

The expansion of the polynomial can be found using the values of

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

^{n}, where n can be any positive integer and x,y are real numbers. Pascal Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement.We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

From the triangle, the seventh line would give the coefficients of the expansion of the polynomial . Thus, we have:

Note:

The expansion of the polynomial can be found using the values of

Maths-

### Use binomial theorem to expand (s^{2} + 3)^{5}

Step 1 of 2:

The given expression is (s

^{2}+ 3)

^{5}, here x = s

^{2}& y = 3 . The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of (s

^{2}+ 3)

^{5}in the binomial equation to get the expansion:

Thus, the expansion is:

### Use binomial theorem to expand (s^{2} + 3)^{5}

Maths-General

Step 1 of 2:

The given expression is (s

^{2}+ 3)

^{5}, here x = s

^{2}& y = 3 . The value of n=5, hence there are 5+1=6 terms in the expressions.

Step 2 of 2:

Substitute the values of (s

^{2}+ 3)

^{5}in the binomial equation to get the expansion:

Thus, the expansion is:

Maths-

### Use polynomial identities to factor the polynomials or simplify the expressions :

ANSWER:

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is:

Note:

Polynomial identities are equations that are true for all possible values of the variable. We can perform polynomial multiplication by applying the distributive property to the multiplication of polynomials.

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is:

Note:

Polynomial identities are equations that are true for all possible values of the variable. We can perform polynomial multiplication by applying the distributive property to the multiplication of polynomials.

### Use polynomial identities to factor the polynomials or simplify the expressions :

Maths-General

ANSWER:

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is:

Note:

Polynomial identities are equations that are true for all possible values of the variable. We can perform polynomial multiplication by applying the distributive property to the multiplication of polynomials.

Hint:

, where a and b can be real values, variables or multiples of both.

We are asked to use polynomial identities to factorize or simplify the expression.

Step 1 of 2:

The given expression is .

Step 2 of 2:

Use the polynomial identity to simplify the expression;

Thus, the factor is:

Note:

Polynomial identities are equations that are true for all possible values of the variable. We can perform polynomial multiplication by applying the distributive property to the multiplication of polynomials.

Maths-

### Use Pascal triangle to expand

ANSWER:

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

From the triangle, the seventh line would give the coefficients of the expansion of the polynomial.

Thus, we have:

Note:

The expansion of the polynomial can be found using the values of

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

^{n}, where n can be any positive integer and x,y are real numbers. Pascal Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement.We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

From the triangle, the seventh line would give the coefficients of the expansion of the polynomial.

Thus, we have:

Note:

The expansion of the polynomial can be found using the values of

### Use Pascal triangle to expand

Maths-General

ANSWER:

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by:

Hint:

Hint:

Pascal's Triangle is a method to know the binomial coefficients of terms of binomial expression (x + y)

^{n}, where n can be any positive integer and x,y are real numbers. Pascal Triangle is represented in a triangular form, it is kind of a number pattern in the form of a triangular arrangement.We are asked to find the binomial expansion of using Pascal’s formula.

Step 1 of 1:

The Pascal’s triangle is given by: