Select All The Expressions That Can Be Factored Into A Squared Binomial.A. $y^2 + 2y + 1$B. $w^2 + 5w + \frac{25}{4}$C. $y^2 - 10y + 5$D. $x^2 - 10x + 25$E. $x^2 + 10x + 25$F. $w^2 + 20w + 40$

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Introduction

In algebra, factoring is a crucial concept that helps us simplify complex expressions and solve equations. One of the most important types of factoring is factoring squared binomials, which are expressions that can be written in the form of (a+b)2(a + b)^2 or (a−b)2(a - b)^2. In this article, we will explore the concept of factoring squared binomials and identify the expressions that can be factored into a squared binomial.

What is a Squared Binomial?

A squared binomial is an expression that can be written in the form of (a+b)2(a + b)^2 or (a−b)2(a - b)^2. This type of expression is also known as a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial.

The Formula for Factoring Squared Binomials

The formula for factoring squared binomials is:

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

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

Using this formula, we can identify the expressions that can be factored into a squared binomial.

Identifying Expressions that Can be Factored into a Squared Binomial

Let's examine each of the expressions given in the problem and determine if they can be factored into a squared binomial.

A. y2+2y+1y^2 + 2y + 1

This expression can be factored into a squared binomial using the formula:

(y+1)2=y2+2y+1(y + 1)^2 = y^2 + 2y + 1

Therefore, expression A can be factored into a squared binomial.

B. w2+5w+254w^2 + 5w + \frac{25}{4}

This expression cannot be factored into a squared binomial because the constant term is not a perfect square.

C. y2−10y+5y^2 - 10y + 5

This expression cannot be factored into a squared binomial because the constant term is not a perfect square.

D. x2−10x+25x^2 - 10x + 25

This expression can be factored into a squared binomial using the formula:

(x−5)2=x2−10x+25(x - 5)^2 = x^2 - 10x + 25

Therefore, expression D can be factored into a squared binomial.

E. x2+10x+25x^2 + 10x + 25

This expression can be factored into a squared binomial using the formula:

(x+5)2=x2+10x+25(x + 5)^2 = x^2 + 10x + 25

Therefore, expression E can be factored into a squared binomial.

F. w2+20w+40w^2 + 20w + 40

This expression cannot be factored into a squared binomial because the constant term is not a perfect square.

Conclusion

In conclusion, expressions A and D can be factored into a squared binomial, while expressions B, C, E, and F cannot be factored into a squared binomial. By using the formula for factoring squared binomials, we can identify the expressions that can be factored into a squared binomial.

Key Takeaways

  • A squared binomial is an expression that can be written in the form of (a+b)2(a + b)^2 or (a−b)2(a - b)^2.
  • The formula for factoring squared binomials is (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 and (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
  • Expressions A and D can be factored into a squared binomial, while expressions B, C, E, and F cannot be factored into a squared binomial.

Practice Problems

  1. Factor the expression (x+3)2(x + 3)^2.
  2. Factor the expression (y−2)2(y - 2)^2.
  3. Determine if the expression x2+6x+9x^2 + 6x + 9 can be factored into a squared binomial.
  4. Determine if the expression y2−8y+16y^2 - 8y + 16 can be factored into a squared binomial.

Answer Key

  1. (x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9
  2. (y−2)2=y2−4y+4(y - 2)^2 = y^2 - 4y + 4
  3. Yes, the expression x2+6x+9x^2 + 6x + 9 can be factored into a squared binomial.
  4. Yes, the expression y2−8y+16y^2 - 8y + 16 can be factored into a squared binomial.
    Factoring Squared Binomials: A Guide to Identifying Expressions ===========================================================

Q&A: Factoring Squared Binomials

Q: What is a squared binomial?

A: A squared binomial is an expression that can be written in the form of (a+b)2(a + b)^2 or (a−b)2(a - b)^2. This type of expression is also known as a perfect square trinomial.

Q: What is the formula for factoring squared binomials?

A: The formula for factoring squared binomials is:

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

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

Q: How do I determine if an expression can be factored into a squared binomial?

A: To determine if an expression can be factored into a squared binomial, you need to check if the expression can be written in the form of (a+b)2(a + b)^2 or (a−b)2(a - b)^2. You can do this by looking at the expression and seeing if it can be factored into a perfect square trinomial.

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

A: Some common mistakes to avoid when factoring squared binomials include:

  • Not recognizing that an expression is a perfect square trinomial
  • Not using the correct formula for factoring squared binomials
  • Not checking if the expression can be factored into a squared binomial before trying to factor it

Q: Can you give an example of how to factor a squared binomial?

A: Let's say we want to factor the expression (x+3)2(x + 3)^2. Using the formula for factoring squared binomials, we get:

(x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9

Q: Can you give an example of how to determine if an expression can be factored into a squared binomial?

A: Let's say we want to determine if the expression x2+6x+9x^2 + 6x + 9 can be factored into a squared binomial. We can check if the expression can be written in the form of (a+b)2(a + b)^2 or (a−b)2(a - b)^2. In this case, we can see that the expression can be written as:

(x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9

Therefore, the expression x2+6x+9x^2 + 6x + 9 can be factored into a squared binomial.

Q: What are some real-world applications of factoring squared binomials?

A: Factoring squared binomials has many real-world applications, including:

  • Simplifying complex expressions in algebra and calculus
  • Solving equations and inequalities in algebra and calculus
  • Finding the area and perimeter of shapes in geometry
  • Modeling real-world phenomena in physics and engineering

Q: Can you give some practice problems for factoring squared binomials?

A: Here are some practice problems for factoring squared binomials:

  1. Factor the expression (x+2)2(x + 2)^2.
  2. Factor the expression (y−3)2(y - 3)^2.
  3. Determine if the expression x2+8x+16x^2 + 8x + 16 can be factored into a squared binomial.
  4. Determine if the expression y2−12y+36y^2 - 12y + 36 can be factored into a squared binomial.

Answer Key

  1. (x+2)2=x2+4x+4(x + 2)^2 = x^2 + 4x + 4
  2. (y−3)2=y2−6y+9(y - 3)^2 = y^2 - 6y + 9
  3. Yes, the expression x2+8x+16x^2 + 8x + 16 can be factored into a squared binomial.
  4. Yes, the expression y2−12y+36y^2 - 12y + 36 can be factored into a squared binomial.