Which Of The Following Represents The Factorization Of The Trinomial Below? X 2 − 14 X + 49 X^2 - 14x + 49 X 2 − 14 X + 49 A. ( X + 7 ) 2 (x+7)^2 ( X + 7 ) 2 B. ( X − 7 ) ( X + 7 (x-7)(x+7 ( X − 7 ) ( X + 7 ]C. ( X − 7 ) 2 (x-7)^2 ( X − 7 ) 2 D. ( X + 2 ) ( X + 7 (x+2)(x+7 ( X + 2 ) ( X + 7 ]

Mar 8, 2025 by ADMIN 294 views

**Factoring Trinomials: A Step-by-Step Guide**

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Understanding Trinomials

A trinomial is a polynomial expression consisting of three terms. It can be written in the form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. Factoring trinomials involves expressing them as a product of simpler expressions, such as binomials or other polynomials.

The Importance of Factoring Trinomials

Factoring trinomials is a crucial skill in algebra, as it allows us to simplify complex expressions, solve equations, and identify the roots of a polynomial. By factoring a trinomial, we can rewrite it in a more manageable form, making it easier to perform operations and analyze the behavior of the polynomial.

The Process of Factoring Trinomials

To factor a trinomial, we need to identify the values of $a$ , $b$ , and $c$ and determine the type of factoring required. There are several methods for factoring trinomials, including:

Factoring by Grouping: This method involves grouping the terms of the trinomial into pairs and factoring out common factors.
Factoring by Difference of Squares: This method involves recognizing that the trinomial can be expressed as the difference of two squares.
Factoring by Perfect Square Trinomials: This method involves recognizing that the trinomial can be expressed as a perfect square trinomial.

Factoring the Given Trinomial

The given trinomial is $x^2 - 14x + 49$ . To factor this trinomial, we need to identify the values of $a$ , $b$ , and $c$ .

Identifying the Values of $a$ , $b$ , and $c$
- $a = 1$
- $b = -14$
- $c = 49$

Determining the Type of Factoring Required

Based on the values of $a$ , $b$ , and $c$ , we can determine the type of factoring required. In this case, the trinomial can be expressed as a perfect square trinomial.

Factoring the Trinomial

To factor the trinomial, we need to recognize that it can be expressed as a perfect square trinomial. A perfect square trinomial can be written in the form of $(x + d)^2$ , where $d$ is a constant.

Recognizing the Perfect Square Trinomial
- The trinomial $x^2 - 14x + 49$ can be expressed as $(x - 7)^2$ .

Verifying the Factored Form

To verify the factored form, we need to multiply the factors together and check if we get the original trinomial.

Multiplying the Factors
- $(x - 7)(x - 7) = x^2 - 14x + 49$

Conclusion

In conclusion, the factored form of the given trinomial is $(x - 7)^2$ . This can be verified by multiplying the factors together and checking if we get the original trinomial.

Answer

The correct answer is C. $(x-7)^2$ .

Additional Examples

Here are some additional examples of factoring trinomials:

Example 1

Factor the trinomial $x^2 + 6x + 9$ .

Identifying the Values of $a$ , $b$ , and $c$
- $a = 1$
- $b = 6$
- $c = 9$
Determining the Type of Factoring Required
- The trinomial can be expressed as a perfect square trinomial.
Factoring the Trinomial
- The trinomial $x^2 + 6x + 9$ can be expressed as $(x + 3)^2$ .

Example 2

Factor the trinomial $x^2 - 8x + 16$ .

Identifying the Values of $a$ , $b$ , and $c$
- $a = 1$
- $b = -8$
- $c = 16$
Determining the Type of Factoring Required
- The trinomial can be expressed as a perfect square trinomial.
Factoring the Trinomial
- The trinomial $x^2 - 8x + 16$ can be expressed as $(x - 4)^2$ .

Example 3

Factor the trinomial $x^2 + 10x + 25$ .

Identifying the Values of $a$ , $b$ , and $c$
- $a = 1$
- $b = 10$
- $c = 25$
Determining the Type of Factoring Required
- The trinomial can be expressed as a perfect square trinomial.
Factoring the Trinomial
- The trinomial $x^2 + 10x + 25$ can be expressed as $(x + 5)^2$ .

Tips and Tricks

Here are some tips and tricks for factoring trinomials:

Use the Perfect Square Trinomial Formula
- The perfect square trinomial formula is $(x + d)^2 = x^2 + 2dx + d^2$ .
Recognize the Difference of Squares
- The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ .
Use the Factoring by Grouping Method
- The factoring by grouping method involves grouping the terms of the trinomial into pairs and factoring out common factors.

Conclusion

In conclusion, factoring trinomials is a crucial skill in algebra that involves expressing a trinomial as a product of simpler expressions. By recognizing the type of factoring required and using the appropriate method, we can factor trinomials and simplify complex expressions.

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Q: What is a trinomial?

A trinomial is a polynomial expression consisting of three terms. It can be written in the form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: Why is factoring trinomials important?

Q: What are the different methods for factoring trinomials?

There are several methods for factoring trinomials, including:

Factoring by Grouping: This method involves grouping the terms of the trinomial into pairs and factoring out common factors.
Factoring by Difference of Squares: This method involves recognizing that the trinomial can be expressed as the difference of two squares.
Factoring by Perfect Square Trinomials: This method involves recognizing that the trinomial can be expressed as a perfect square trinomial.

Q: How do I determine the type of factoring required for a trinomial?

To determine the type of factoring required for a trinomial, we need to identify the values of $a$ , $b$ , and $c$ and examine the trinomial for patterns or relationships that suggest a particular factoring method.

Q: What is a perfect square trinomial?

A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. It has the form $(x + d)^2 = x^2 + 2dx + d^2$ , where $d$ is a constant.

Q: How do I factor a perfect square trinomial?

To factor a perfect square trinomial, we need to recognize the pattern and identify the binomial that, when squared, gives the trinomial.

Q: What is the difference of squares formula?

The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ .

Q: How do I factor a trinomial using the difference of squares formula?

To factor a trinomial using the difference of squares formula, we need to recognize that the trinomial can be expressed as the difference of two squares and then apply the formula.

Q: What is factoring by grouping?

Factoring by grouping involves grouping the terms of the trinomial into pairs and factoring out common factors.

Q: How do I factor a trinomial using factoring by grouping?

To factor a trinomial using factoring by grouping, we need to group the terms of the trinomial into pairs and then factor out common factors.

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

Some common mistakes to avoid when factoring trinomials include:

Not recognizing the type of factoring required
Not identifying the values of $a$ , $b$ , and $c$
Not examining the trinomial for patterns or relationships
Not recognizing the difference of squares or perfect square trinomial patterns

Q: How can I practice factoring trinomials?

To practice factoring trinomials, we can try the following:

Work through examples and exercises
Use online resources and practice problems
Seek help from a teacher or tutor
Join a study group or online community

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

Some real-world applications of factoring trinomials include:

Solving equations and inequalities
Analyzing the behavior of polynomials
Modeling real-world phenomena
Optimizing systems and processes