Which Of The Binomials Below Is A Factor Of This Trinomial?$3x^2 - 21x + 24$A. $x - 1$ B. $x + 4$ C. $x + 1$ D. $x - 4$

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Introduction

In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. This is a crucial concept in mathematics, as it allows us to solve equations and inequalities more efficiently. In this article, we will explore the concept of factoring and determine which of the given binomials is a factor of the trinomial 3x2−21x+243x^2 - 21x + 24.

What is Factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials. For example, the polynomial x2+5x+6x^2 + 5x + 6 can be factored as (x+3)(x+2)(x + 3)(x + 2). Factoring is an essential tool in algebra, as it allows us to solve equations and inequalities more efficiently.

Types of Factoring

There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves factoring out the greatest common factor of the terms in the polynomial.
  • Difference of Squares Factoring: This involves factoring the difference of two squares, such as a2−b2a^2 - b^2.
  • Sum and Difference Factoring: This involves factoring the sum or difference of two terms, such as a+ba + b or a−ba - b.
  • Quadratic Formula Factoring: This involves factoring a quadratic expression using the quadratic formula.

Factoring the Trinomial

To determine which of the given binomials is a factor of the trinomial 3x2−21x+243x^2 - 21x + 24, we need to factor the trinomial. We can start by factoring out the greatest common factor, which is 3.

3x^2 - 21x + 24 = 3(x^2 - 7x + 8)

Next, we can try to factor the quadratic expression x2−7x+8x^2 - 7x + 8. We can use the quadratic formula to find the roots of the quadratic expression.

x^2 - 7x + 8 = (x - 4)(x - 2)

Therefore, the trinomial 3x2−21x+243x^2 - 21x + 24 can be factored as 3(x−4)(x−2)3(x - 4)(x - 2).

Which Binomial is a Factor?

Now that we have factored the trinomial, we can determine which of the given binomials is a factor. We can see that the binomial x−4x - 4 is a factor of the trinomial, as it is one of the factors of the factored form 3(x−4)(x−2)3(x - 4)(x - 2).

Conclusion

In conclusion, the binomial x−4x - 4 is a factor of the trinomial 3x2−21x+243x^2 - 21x + 24. This is because the trinomial can be factored as 3(x−4)(x−2)3(x - 4)(x - 2), and the binomial x−4x - 4 is one of the factors.

Answer

The answer is D. x−4x - 4.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Additional Resources

  • Khan Academy: Algebra
  • MIT OpenCourseWare: Algebra
  • Wolfram Alpha: Algebra

Introduction

In our previous article, we explored the concept of factoring and determined which of the given binomials is a factor of the trinomial 3x2−21x+243x^2 - 21x + 24. In this article, we will answer some frequently asked questions about factoring and algebra.

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: How do I determine if a binomial is a factor of a trinomial?

A: To determine if a binomial is a factor of a trinomial, you can try to factor the trinomial and see if the binomial is one of the factors. You can also use the quadratic formula to find the roots of the quadratic expression and see if the binomial is a factor.

Q: What is the greatest common factor (GCF) of a polynomial?

A: The greatest common factor (GCF) of a polynomial is the largest polynomial that divides each term of the polynomial without leaving a remainder.

Q: How do I factor a quadratic expression using the quadratic formula?

A: To factor a quadratic expression using the quadratic formula, you can use the formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic expression.

Q: What is the difference between a monomial and a polynomial?

A: A monomial is a single term that consists of a coefficient and a variable, while a polynomial is an expression that consists of two or more terms.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you can combine like terms by adding or subtracting the coefficients of the terms.

Q: What is the order of operations in algebra?

A: The order of operations in algebra is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Isolate the variable by adding or subtracting the same value to both sides of the equation.
  2. Use inverse operations to isolate the variable.
  3. Check your solution by plugging it back into the original equation.

Conclusion

In conclusion, factoring and algebra are essential concepts in mathematics that can be used to solve equations and inequalities. By understanding the concepts of factoring and algebra, you can solve a wide range of problems in mathematics and science.

Additional Resources

  • Khan Academy: Algebra
  • MIT OpenCourseWare: Algebra
  • Wolfram Alpha: Algebra

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Note: The references and additional resources provided are for informational purposes only and are not directly related to the content of this article.