Which Of The Following Is The Correct Factorization Of This Trinomial?$4x^2 - 0x - 25$A. \[$(4x - 5)(x + 5)\$\]B. \[$(4x - 1)(x + 25)\$\]C. \[$(2x - 5)(2x + 5)\$\]D. \[$(2x - 1)(2x + 25)\$\]

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Introduction

Factoring trinomials is a fundamental concept in algebra that can be used to simplify complex expressions and solve equations. In this article, we will explore the correct factorization of a given trinomial and provide a step-by-step guide on how to factor trinomials.

What is a Trinomial?

A trinomial is a polynomial expression that consists of three terms. It can be written in the form of ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, and xx is the variable.

The Given Trinomial

The given trinomial is 4x2−0x−254x^2 - 0x - 25. To factor this trinomial, we need to find two binomials whose product is equal to the given trinomial.

Factoring the Trinomial

To factor the trinomial, we need to find two binomials whose product is equal to the given trinomial. We can start by looking for two numbers whose product is equal to the product of the coefficients of the two middle terms, which is −25-25. These numbers are 55 and −5-5.

Option A: (4x−5)(x+5)(4x - 5)(x + 5)

Let's start by multiplying the two binomials:

(4x−5)(x+5)=4x2+20x−5x−25(4x - 5)(x + 5) = 4x^2 + 20x - 5x - 25

Combine like terms:

4x2+15x−254x^2 + 15x - 25

This is not equal to the given trinomial, so option A is incorrect.

Option B: (4x−1)(x+25)(4x - 1)(x + 25)

Let's start by multiplying the two binomials:

(4x−1)(x+25)=4x2+100x−x−25(4x - 1)(x + 25) = 4x^2 + 100x - x - 25

Combine like terms:

4x2+99x−254x^2 + 99x - 25

This is not equal to the given trinomial, so option B is incorrect.

Option C: (2x−5)(2x+5)(2x - 5)(2x + 5)

Let's start by multiplying the two binomials:

(2x−5)(2x+5)=4x2+10x−10x−25(2x - 5)(2x + 5) = 4x^2 + 10x - 10x - 25

Combine like terms:

4x2−254x^2 - 25

This is equal to the given trinomial, so option C is correct.

Option D: (2x−1)(2x+25)(2x - 1)(2x + 25)

Let's start by multiplying the two binomials:

(2x−1)(2x+25)=4x2+50x−2x−25(2x - 1)(2x + 25) = 4x^2 + 50x - 2x - 25

Combine like terms:

4x2+48x−254x^2 + 48x - 25

This is not equal to the given trinomial, so option D is incorrect.

Conclusion

In conclusion, the correct factorization of the given trinomial is (2x−5)(2x+5)(2x - 5)(2x + 5). This is option C.

Tips and Tricks

  • When factoring trinomials, look for two numbers whose product is equal to the product of the coefficients of the two middle terms.
  • Use the distributive property to multiply the two binomials.
  • Combine like terms to simplify the expression.

Practice Problems

  • Factor the trinomial 9x2+12x+49x^2 + 12x + 4.
  • Factor the trinomial 16x2−12x−2016x^2 - 12x - 20.

References

Glossary

  • Trinomial: A polynomial expression that consists of three terms.
  • Binomial: A polynomial expression that consists of two terms.
  • Distributive property: A property of arithmetic that states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.

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Introduction

Factoring trinomials is a fundamental concept in algebra that can be used to simplify complex expressions and solve equations. In this article, we will provide a Q&A guide on factoring trinomials, covering common questions and topics.

Q: What is a trinomial?

A: A trinomial is a polynomial expression that consists of three terms. It can be written in the form of ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I factor a trinomial?

A: To factor a trinomial, you need to find two binomials whose product is equal to the given trinomial. You can start by looking for two numbers whose product is equal to the product of the coefficients of the two middle terms.

Q: What are the common mistakes when factoring trinomials?

A: Some common mistakes when factoring trinomials include:

  • Not looking for two numbers whose product is equal to the product of the coefficients of the two middle terms.
  • Not using the distributive property to multiply the two binomials.
  • Not combining like terms to simplify the expression.

Q: How do I know which binomial to multiply first?

A: To determine which binomial to multiply first, you can look at the signs of the coefficients of the two middle terms. If the signs are the same, you can multiply the binomials in either order. If the signs are different, you need to multiply the binomials in the order that will give you the correct signs.

Q: What is the difference between factoring and simplifying?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while simplifying involves combining like terms to reduce the complexity of an expression.

Q: Can I factor a trinomial that has a negative coefficient?

A: Yes, you can factor a trinomial that has a negative coefficient. You can start by looking for two numbers whose product is equal to the product of the coefficients of the two middle terms, and then use the distributive property to multiply the two binomials.

Q: How do I factor a trinomial that has a zero coefficient?

A: If a trinomial has a zero coefficient, it means that one of the terms is equal to zero. In this case, you can factor out the zero term and simplify the expression.

Q: Can I use a calculator to factor a trinomial?

A: Yes, you can use a calculator to factor a trinomial. However, it's always a good idea to check your work by multiplying the two binomials and simplifying the expression.

Q: How do I know if a trinomial can be factored?

A: A trinomial can be factored if it can be expressed as a product of two binomials. You can use the distributive property to multiply the two binomials and simplify the expression.

Q: What are some common applications of factoring trinomials?

A: Factoring trinomials has many applications in algebra and other areas of mathematics. Some common applications include:

  • Solving quadratic equations
  • Simplifying complex expressions
  • Finding the roots of a polynomial

Conclusion

In conclusion, factoring trinomials is a fundamental concept in algebra that can be used to simplify complex expressions and solve equations. By following the steps outlined in this article, you can factor trinomials and apply the concept to real-world problems.

Practice Problems

  • Factor the trinomial 9x2+12x+49x^2 + 12x + 4.
  • Factor the trinomial 16x2−12x−2016x^2 - 12x - 20.

References

Glossary

  • Trinomial: A polynomial expression that consists of three terms.
  • Binomial: A polynomial expression that consists of two terms.
  • Distributive property: A property of arithmetic that states that the product of a number and a sum is equal to the sum of the products of the number and each term in the sum.