Factor The Expression: D 2 − 10 D + 25 D^2 - 10d + 25 D 2 − 10 D + 25 A. { (d + 5)(d - 5)$}$ B. { (d - 5)(d - 5)$}$ C. { (d - 15)(d + 10)$}$ D. { (d + 5)(d - 20)$}$

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Introduction

Factoring an algebraic expression is a fundamental concept in mathematics that involves expressing the given expression as a product of simpler expressions. In this article, we will focus on factoring the quadratic expression d2−10d+25d^2 - 10d + 25. We will explore the different methods of factoring and provide a step-by-step guide on how to factor the given expression.

What is Factoring?

Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves finding the factors of the given expression, which are the numbers or expressions that multiply together to give the original expression. Factoring is an essential concept in algebra and is used to simplify complex expressions, solve equations, and graph functions.

Methods of Factoring

There are several methods of factoring, including:

  • Factoring by Grouping: This method involves grouping the terms of the expression into pairs and factoring out the common factors from each pair.
  • Factoring by Difference of Squares: This method involves factoring expressions of the form a2−b2a^2 - b^2, where aa and bb are expressions.
  • Factoring by Perfect Square Trinomials: This method involves factoring expressions of the form a2+2ab+b2a^2 + 2ab + b^2, where aa and bb are expressions.

Factoring the Expression d2−10d+25d^2 - 10d + 25

The given expression d2−10d+25d^2 - 10d + 25 is a quadratic expression, which can be factored using the method of factoring by perfect square trinomials.

Step 1: Identify the Perfect Square Trinomial

The expression d2−10d+25d^2 - 10d + 25 can be written as (d)2−2(d)(5)+(5)2(d)^2 - 2(d)(5) + (5)^2, which is a perfect square trinomial.

Step 2: Factor the Perfect Square Trinomial

Using the formula for factoring perfect square trinomials, we can write the expression as (d−5)2(d - 5)^2.

Step 3: Simplify the Expression

The expression (d−5)2(d - 5)^2 can be simplified to (d−5)(d−5)(d - 5)(d - 5).

Conclusion

In conclusion, the expression d2−10d+25d^2 - 10d + 25 can be factored as (d−5)(d−5)(d - 5)(d - 5). This is the final answer to the problem.

Final Answer

The final answer is: (d−5)(d−5)\boxed{(d - 5)(d - 5)}

Discussion

The expression d2−10d+25d^2 - 10d + 25 can be factored using the method of factoring by perfect square trinomials. This method involves identifying the perfect square trinomial and factoring it using the formula. The final answer is (d−5)(d−5)(d - 5)(d - 5).

Related Topics

  • Factoring by Grouping
  • Factoring by Difference of Squares
  • Factoring by Perfect Square Trinomials
  • Quadratic Expressions
  • Algebraic Identities

References

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Introduction

In our previous article, we explored the concept of factoring an algebraic expression and provided a step-by-step guide on how to factor the quadratic expression d2−10d+25d^2 - 10d + 25. In this article, we will provide a Q&A guide to help you better understand the concept of factoring and how to apply it to different types of expressions.

Q&A

Q: What is factoring?

A: Factoring is a process of expressing an algebraic expression as a product of simpler expressions. It involves finding the factors of the given expression, which are the numbers or expressions that multiply together to give the original expression.

Q: What are the different methods of factoring?

A: There are several methods of factoring, including:

  • Factoring by Grouping: This method involves grouping the terms of the expression into pairs and factoring out the common factors from each pair.
  • Factoring by Difference of Squares: This method involves factoring expressions of the form a2−b2a^2 - b^2, where aa and bb are expressions.
  • Factoring by Perfect Square Trinomials: This method involves factoring expressions of the form a2+2ab+b2a^2 + 2ab + b^2, where aa and bb are expressions.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to identify the perfect square trinomial and factor it using the formula. The formula for factoring a perfect square trinomial is:

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Q: What is a perfect square trinomial?

A: A perfect square trinomial is an expression of the form a2+2ab+b2a^2 + 2ab + b^2, where aa and bb are expressions. It can be factored as (a+b)2(a + b)^2.

Q: How do I factor an expression using the method of factoring by grouping?

A: To factor an expression using the method of factoring by grouping, you need to group the terms of the expression into pairs and factor out the common factors from each pair.

Q: What are some common mistakes to avoid when factoring?

A: Some common mistakes to avoid when factoring include:

  • Not identifying the perfect square trinomial: Make sure to identify the perfect square trinomial before factoring.
  • Not factoring out the common factors: Make sure to factor out the common factors from each pair of terms.
  • Not checking the answer: Make sure to check the answer by multiplying the factors together.

Conclusion

In conclusion, factoring is a powerful tool for simplifying complex expressions and solving equations. By understanding the different methods of factoring and how to apply them, you can become proficient in factoring and solve a wide range of problems.

Final Tips

  • Practice, practice, practice: The more you practice factoring, the more comfortable you will become with the different methods and techniques.
  • Use online resources: There are many online resources available that can help you learn and practice factoring, including video tutorials, practice problems, and interactive quizzes.
  • Seek help when needed: Don't be afraid to ask for help if you are struggling with a particular concept or problem.

Related Topics

  • Factoring by Grouping
  • Factoring by Difference of Squares
  • Factoring by Perfect Square Trinomials
  • Quadratic Expressions
  • Algebraic Identities

References