Simplify The Expression: \[$(b+1)(4b-4)\$\]

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. The given expression, (b+1)(4b−4)(b+1)(4b-4), is a product of two binomials, and we need to simplify it using the distributive property. In this article, we will walk through the steps to simplify the expression and provide a clear understanding of the process.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more binomials. It states that for any real numbers aa, bb, and cc, the following equation holds:

a(b+c)=ab+aca(b+c) = ab + ac

This property can be extended to more than two binomials, and it is a powerful tool for simplifying expressions.

Applying the Distributive Property

To simplify the expression (b+1)(4b−4)(b+1)(4b-4), we will apply the distributive property. We will multiply each term in the first binomial, (b+1)(b+1), by each term in the second binomial, (4b−4)(4b-4).

(b+1)(4b−4)=b(4b−4)+1(4b−4)(b+1)(4b-4) = b(4b-4) + 1(4b-4)

Expanding the Terms

Now, we will expand each term in the expression.

b(4b−4)=4b2−4bb(4b-4) = 4b^2 - 4b

1(4b−4)=4b−41(4b-4) = 4b - 4

Combining Like Terms

We can now combine like terms in the expression.

4b2−4b+4b−4=4b2−44b^2 - 4b + 4b - 4 = 4b^2 - 4

Simplifying the Expression

The final step is to simplify the expression by combining the remaining terms.

4b2−4=4(b2−1)4b^2 - 4 = 4(b^2 - 1)

Conclusion

In this article, we simplified the expression (b+1)(4b−4)(b+1)(4b-4) using the distributive property. We applied the property to expand the product of two binomials, combined like terms, and simplified the resulting expression. The final simplified form of the expression is 4(b2−1)4(b^2 - 1).

Final Answer

The final answer is 4(b2−1)\boxed{4(b^2 - 1)}.

Example Use Case

Simplifying expressions like (b+1)(4b−4)(b+1)(4b-4) is an essential skill in algebra. It helps us solve equations and manipulate mathematical statements. For example, if we have an equation like 2x2+3x−1=02x^2 + 3x - 1 = 0, we can use the simplified expression 4(b2−1)4(b^2 - 1) to find the solutions.

Tips and Tricks

  • When simplifying expressions, always look for opportunities to apply the distributive property.
  • Combine like terms whenever possible to simplify the expression.
  • Use the distributive property to expand products of binomials.

Common Mistakes

  • Failing to apply the distributive property when simplifying expressions.
  • Not combining like terms when possible.
  • Not using the distributive property to expand products of binomials.

Frequently Asked Questions

  • What is the distributive property?
    • The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more binomials.
  • How do I simplify expressions using the distributive property?
    • To simplify expressions using the distributive property, apply the property to expand the product of two or more binomials, combine like terms, and simplify the resulting expression.
  • What is the final simplified form of the expression (b+1)(4b−4)(b+1)(4b-4)?
    • The final simplified form of the expression (b+1)(4b−4)(b+1)(4b-4) is 4(b2−1)4(b^2 - 1).

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Introduction

In our previous article, we simplified the expression (b+1)(4b−4)(b+1)(4b-4) using the distributive property. In this article, we will provide a Q&A section to help you better understand the concept and address any questions you may have.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more binomials. It states that for any real numbers aa, bb, and cc, the following equation holds:

a(b+c)=ab+aca(b+c) = ab + ac

Q: How do I simplify expressions using the distributive property?

A: To simplify expressions using the distributive property, apply the property to expand the product of two or more binomials, combine like terms, and simplify the resulting expression.

Q: What is the final simplified form of the expression (b+1)(4b−4)(b+1)(4b-4)?

A: The final simplified form of the expression (b+1)(4b−4)(b+1)(4b-4) is 4(b2−1)4(b^2 - 1).

Q: Can I use the distributive property to simplify expressions with more than two binomials?

A: Yes, you can use the distributive property to simplify expressions with more than two binomials. The property can be extended to more than two binomials, and it is a powerful tool for simplifying expressions.

Q: How do I combine like terms when simplifying expressions?

A: To combine like terms when simplifying expressions, look for terms that have the same variable and coefficient. Then, add or subtract the coefficients of the like terms.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

  • Failing to apply the distributive property when simplifying expressions.
  • Not combining like terms when possible.
  • Not using the distributive property to expand products of binomials.

Q: Can I use the distributive property to simplify expressions with variables and constants?

A: Yes, you can use the distributive property to simplify expressions with variables and constants. The property can be applied to expand products of binomials, regardless of whether the terms are variables or constants.

Q: How do I check my work when simplifying expressions?

A: To check your work when simplifying expressions, plug the simplified expression back into the original equation and verify that it is true. This will help you ensure that your simplification is correct.

Example Use Cases

Example 1: Simplifying an Expression with Two Binomials

Suppose we have the expression (x+2)(3x−1)(x+2)(3x-1). We can use the distributive property to simplify this expression:

(x+2)(3x−1)=x(3x−1)+2(3x−1)(x+2)(3x-1) = x(3x-1) + 2(3x-1)

Expanding the terms, we get:

3x2−x+6x−23x^2 - x + 6x - 2

Combining like terms, we get:

3x2+5x−23x^2 + 5x - 2

Example 2: Simplifying an Expression with More Than Two Binomials

Suppose we have the expression (x+2)(3x−1)(x+4)(x+2)(3x-1)(x+4). We can use the distributive property to simplify this expression:

(x+2)(3x−1)(x+4)=(x+2)(3x2+11x−4)(x+2)(3x-1)(x+4) = (x+2)(3x^2 + 11x - 4)

Expanding the terms, we get:

3x3+11x2−4x+6x2+22x−83x^3 + 11x^2 - 4x + 6x^2 + 22x - 8

Combining like terms, we get:

3x3+17x2+18x−83x^3 + 17x^2 + 18x - 8

Tips and Tricks

  • When simplifying expressions, always look for opportunities to apply the distributive property.
  • Combine like terms whenever possible to simplify the expression.
  • Use the distributive property to expand products of binomials.
  • Check your work by plugging the simplified expression back into the original equation.

Common Mistakes

  • Failing to apply the distributive property when simplifying expressions.
  • Not combining like terms when possible.
  • Not using the distributive property to expand products of binomials.

Frequently Asked Questions

  • What is the distributive property?
    • The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more binomials.
  • How do I simplify expressions using the distributive property?
    • To simplify expressions using the distributive property, apply the property to expand the product of two or more binomials, combine like terms, and simplify the resulting expression.
  • What is the final simplified form of the expression (b+1)(4b−4)(b+1)(4b-4)?
    • The final simplified form of the expression (b+1)(4b−4)(b+1)(4b-4) is 4(b2−1)4(b^2 - 1).