After Reading The Selection, Please Expand The Following Without Multiplying The Binomials:a) { (x+3)^4$}$b) { (x-y)^5$}$c) { (a+b)^7$}$d) { (x+4)^9$}$e) { (y-2x)^6$}$f) { (3x+1)^3$}$

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Introduction

In algebra, binomials are expressions consisting of two terms. Expanding binomials is a crucial concept in mathematics, as it allows us to simplify and manipulate expressions involving binomials. In this article, we will explore the process of expanding binomials, focusing on the multiplication of binomials raised to various powers.

The Binomial Theorem

The binomial theorem is a fundamental concept in algebra that provides a formula for expanding binomials raised to a power. The theorem states that for any positive integer n, the expansion of (a + b)^n is given by:

(a + b)^n = ∑[k=0 to n] (n choose k) * a^(n-k) * b^k

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items from a set of n items.

Expanding Binomials

Now that we have a solid understanding of the binomial theorem, let's apply it to expand the given binomials.

a) (x + 3)^4

To expand (x + 3)^4, we will use the binomial theorem with a = x, b = 3, and n = 4.

(x + 3)^4 = ∑[k=0 to 4] (4 choose k) * x^(4-k) * 3^k

Expanding the expression, we get:

(x + 3)^4 = x^4 + 4x^3(3) + 6x2(3)2 + 4x(3)^3 + (3)^4

Simplifying the expression, we get:

(x + 3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81

b) (x - y)^5

To expand (x - y)^5, we will use the binomial theorem with a = x, b = -y, and n = 5.

(x - y)^5 = ∑[k=0 to 5] (5 choose k) * x^(5-k) * (-y)^k

Expanding the expression, we get:

(x - y)^5 = x^5 + 5x^4(-y) + 10x3(-y)2 + 10x2(-y)3 + 5x(-y)^4 + (-y)^5

Simplifying the expression, we get:

(x - y)^5 = x^5 - 5x^4y + 10x3y2 - 10x2y3 + 5xy^4 - y^5

c) (a + b)^7

To expand (a + b)^7, we will use the binomial theorem with a = a, b = b, and n = 7.

(a + b)^7 = ∑[k=0 to 7] (7 choose k) * a^(7-k) * b^k

Expanding the expression, we get:

(a + b)^7 = a^7 + 7a^6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab^6 + b^7

d) (x + 4)^9

To expand (x + 4)^9, we will use the binomial theorem with a = x, b = 4, and n = 9.

(x + 4)^9 = ∑[k=0 to 9] (9 choose k) * x^(9-k) * 4^k

Expanding the expression, we get:

(x + 4)^9 = x^9 + 9x^8(4) + 36x7(4)2 + 84x6(4)3 + 126x5(4)4 + 126x4(4)5 + 84x3(4)6 + 36x2(4)7 + 9x(4)^8 + (4)^9

Simplifying the expression, we get:

(x + 4)^9 = x^9 + 36x^8 + 576x^7 + 1344x^6 + 2688x^5 + 4032x^4 + 4032x^3 + 2304x^2 + 9216x + 262144

e) (y - 2x)^6

To expand (y - 2x)^6, we will use the binomial theorem with a = y, b = -2x, and n = 6.

(y - 2x)^6 = ∑[k=0 to 6] (6 choose k) * y^(6-k) * (-2x)^k

Expanding the expression, we get:

(y - 2x)^6 = y^6 + 6y^5(-2x) + 15y4(-2x)2 + 20y3(-2x)3 + 15y2(-2x)4 + 6y(-2x)^5 + (-2x)^6

Simplifying the expression, we get:

(y - 2x)^6 = y^6 - 12xy^5 + 60x2y4 - 160x3y3 + 240x4y2 - 192x^5y + 64x^6

f) (3x + 1)^3

To expand (3x + 1)^3, we will use the binomial theorem with a = 3x, b = 1, and n = 3.

(3x + 1)^3 = ∑[k=0 to 3] (3 choose k) * (3x)^(3-k) * 1^k

Expanding the expression, we get:

(3x + 1)^3 = (3x)^3 + 3(3x)^2(1) + 3(3x)(1)^2 + 1^3

Simplifying the expression, we get:

(3x + 1)^3 = 27x^3 + 27x^2 + 9x + 1

Conclusion

In this article, we have explored the process of expanding binomials using the binomial theorem. We have applied the theorem to expand various binomials raised to different powers. By following the steps outlined in this article, you should be able to expand binomials with ease. Remember to use the binomial theorem to simplify and manipulate expressions involving binomials.

Practice Problems

  1. Expand (x + 2)^5
  2. Expand (y - 3)^4
  3. Expand (a + b)^6
  4. Expand (x + 5)^8
  5. Expand (y - 2x)^5

Answer Key

  1. x^5 + 5x^4(2) + 10x3(2)2 + 10x2(2)3 + 5x(2)^4 + (2)^5
  2. y^4 + 4y^3(-3) + 6y2(-3)2 + 4y(-3)^3 + (-3)^4
  3. a^6 + 6a^5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab^5 + b^6
  4. x^8 + 8x^7(5) + 28x6(5)2 + 56x5(5)3 + 70x4(5)4 + 56x3(5)5 + 28x2(5)6 + 8x(5)^7 + (5)^8
  5. y^5 - 5y^4(2x) + 10y3(2x)2 - 10y2(2x)3 + 5y(2x)^4 - (2x)^5
    Expanding Binomials: A Q&A Guide =====================================

Introduction

In our previous article, we explored the process of expanding binomials using the binomial theorem. In this article, we will answer some of the most frequently asked questions about expanding binomials.

Q: What is the binomial theorem?

A: The binomial theorem is a formula for expanding binomials raised to a power. It states that for any positive integer n, the expansion of (a + b)^n is given by:

(a + b)^n = ∑[k=0 to n] (n choose k) * a^(n-k) * b^k

Q: How do I apply the binomial theorem to expand a binomial?

A: To apply the binomial theorem, you need to follow these steps:

  1. Identify the binomial and the power to which it is raised.
  2. Determine the values of a and b.
  3. Use the binomial theorem formula to expand the binomial.

Q: What is the binomial coefficient?

A: The binomial coefficient, denoted by (n choose k), is the number of ways to choose k items from a set of n items. It is calculated using the formula:

(n choose k) = n! / (k!(n-k)!)

Q: How do I calculate the binomial coefficient?

A: To calculate the binomial coefficient, you can use the formula:

(n choose k) = n! / (k!(n-k)!)

For example, to calculate (5 choose 2), you would use the formula:

(5 choose 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4 × 3 × 2 × 1) / ((2 × 1)(3 × 2 × 1)) = 120 / (2 × 6) = 120 / 12 = 10

Q: What is the difference between expanding a binomial and multiplying two binomials?

A: Expanding a binomial involves using the binomial theorem to simplify an expression involving a binomial raised to a power. Multiplying two binomials involves using the distributive property to multiply each term in one binomial by each term in the other binomial.

Q: How do I multiply two binomials?

A: To multiply two binomials, you can use the distributive property to multiply each term in one binomial by each term in the other binomial. For example, to multiply (x + 2) and (x + 3), you would use the distributive property to get:

(x + 2)(x + 3) = x(x + 3) + 2(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6

Q: Can I use the binomial theorem to expand a binomial with a negative exponent?

A: Yes, you can use the binomial theorem to expand a binomial with a negative exponent. To do this, you need to rewrite the binomial with a positive exponent and then apply the binomial theorem.

Q: Can I use the binomial theorem to expand a binomial with a fractional exponent?

A: Yes, you can use the binomial theorem to expand a binomial with a fractional exponent. To do this, you need to rewrite the binomial with a positive exponent and then apply the binomial theorem.

Conclusion

In this article, we have answered some of the most frequently asked questions about expanding binomials. We hope that this article has been helpful in clarifying any confusion you may have had about expanding binomials. If you have any further questions, please don't hesitate to ask.

Practice Problems

  1. Expand (x + 2)^5
  2. Expand (y - 3)^4
  3. Expand (a + b)^6
  4. Expand (x + 5)^8
  5. Expand (y - 2x)^5

Answer Key

  1. x^5 + 5x^4(2) + 10x3(2)2 + 10x2(2)3 + 5x(2)^4 + (2)^5
  2. y^4 + 4y^3(-3) + 6y2(-3)2 + 4y(-3)^3 + (-3)^4
  3. a^6 + 6a^5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab^5 + b^6
  4. x^8 + 8x^7(5) + 28x6(5)2 + 56x5(5)3 + 70x4(5)4 + 56x3(5)5 + 28x2(5)6 + 8x(5)^7 + (5)^8
  5. y^5 - 5y^4(2x) + 10y3(2x)2 - 10y2(2x)3 + 5y(2x)^4 - (2x)^5