What Is $12x^3 - 9x^2 - 4x + 3$ In Factored Form? ( □ X 2 − □ ) ( □ X − □ (\square X^2 - \square)(\square X - \square ( □ X 2 − □ ) ( □ X − □ ]

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Introduction to Factoring Polynomials

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and simplifying expressions. In this article, we will focus on factoring a given polynomial, $12x^3 - 9x^2 - 4x + 3$, and explore the different techniques used to achieve this.

Understanding the Polynomial

Before we begin factoring, it's essential to understand the structure of the given polynomial. The polynomial is a cubic expression, meaning it has a degree of 3. It consists of four terms: $12x^3$, $-9x^2$, $-4x$, and $3$. Our goal is to express this polynomial as a product of two binomials, each of the form $(ax^2 + bx + c)$.

Factoring by Grouping

One of the techniques used to factor polynomials is factoring by grouping. This method involves grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair. To factor the given polynomial using this method, we can start by grouping the first two terms and the last two terms.

Step 1: Group the first two terms

12x3−9x2=3x2(4x−3)12x^3 - 9x^2 = 3x^2(4x - 3)

Step 2: Group the last two terms

−4x+3=−(4x−3)-4x + 3 = -(4x - 3)

Step 3: Factor out the GCF from each pair

Now that we have grouped the terms, we can factor out the GCF from each pair. The GCF of the first pair is $3x^2$, and the GCF of the second pair is $-(4x - 3)$.

12x3−9x2−4x+3=3x2(4x−3)−(4x−3)12x^3 - 9x^2 - 4x + 3 = 3x^2(4x - 3) - (4x - 3)

Step 4: Factor out the common binomial

Now that we have factored out the GCF from each pair, we can factor out the common binomial $(4x - 3)$.

12x3−9x2−4x+3=(3x2−1)(4x−3)12x^3 - 9x^2 - 4x + 3 = (3x^2 - 1)(4x - 3)

Factoring by Greatest Common Factor (GCF)

Another technique used to factor polynomials is factoring by GCF. This method involves factoring out the greatest common factor from all the terms of the polynomial. To factor the given polynomial using this method, we can start by identifying the GCF of all the terms.

Step 1: Identify the GCF

The GCF of the polynomial is 1, since there is no common factor among all the terms.

Step 2: Factor out the GCF

Since the GCF is 1, we cannot factor out any common factor from the polynomial.

Factoring by Synthetic Division

Synthetic division is a technique used to factor polynomials by dividing the polynomial by a linear factor. To factor the given polynomial using this method, we can start by identifying a linear factor that divides the polynomial.

Step 1: Identify a linear factor

One possible linear factor is $(x - 1)$.

Step 2: Perform synthetic division

Performing synthetic division with the linear factor $(x - 1)$, we get:

112−9−4312−53123−16\begin{array}{c|rrrr} 1 & 12 & -9 & -4 & 3 \\ & & 12 & -5 & 3 \\ \hline & 12 & 3 & -1 & 6 \end{array}

Step 3: Factor the quotient

The quotient is $12x^2 + 3x - 1$.

Step 4: Factor the quotient further

We can factor the quotient further by factoring out the GCF.

12x2+3x−1=(3x−1)(4x+1)12x^2 + 3x - 1 = (3x - 1)(4x + 1)

Conclusion

In this article, we have explored different techniques used to factor the polynomial $12x^3 - 9x^2 - 4x + 3$. We have used factoring by grouping, factoring by GCF, and synthetic division to factor the polynomial. The final factored form of the polynomial is $(3x - 1)(4x + 1)(x + 1)$. Factoring polynomials is an essential concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By mastering the techniques used to factor polynomials, we can solve equations, find roots, and simplify expressions.

Final Answer

The final factored form of the polynomial $12x^3 - 9x^2 - 4x + 3$ is:

(3x - 1)(4x + 1)(x + 1)$<br/> # Q&A: Factoring Polynomials

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we explored different techniques used to factor the polynomial $12x^3 - 9x^2 - 4x + 3$. In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor polynomials by grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair.

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

A: To factor a polynomial using factoring by grouping, you can start by grouping the first two terms and the last two terms. Then, factor out the GCF from each pair. Finally, factor out the common binomial from the two pairs.

Q: What is factoring by GCF?

A: Factoring by GCF is a technique used to factor polynomials by factoring out the greatest common factor from all the terms of the polynomial.

Q: How do I factor a polynomial using factoring by GCF?

A: To factor a polynomial using factoring by GCF, you can start by identifying the GCF of all the terms. If the GCF is 1, you cannot factor out any common factor from the polynomial.

Q: What is synthetic division?

A: Synthetic division is a technique used to factor polynomials by dividing the polynomial by a linear factor.

Q: How do I factor a polynomial using synthetic division?

A: To factor a polynomial using synthetic division, you can start by identifying a linear factor that divides the polynomial. Then, perform synthetic division with the linear factor. Finally, factor the quotient further by factoring out the GCF.

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

A: Some common mistakes to avoid when factoring polynomials include:

  • Factoring out a term that is not a common factor
  • Factoring out a term that is not a factor of the polynomial
  • Not factoring out the greatest common factor
  • Not factoring out the common binomial

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

A: A polynomial can be factored if it has a greatest common factor (GCF) that can be factored out. Additionally, a polynomial can be factored if it has a linear factor that divides the polynomial.

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

A: Factoring polynomials has many real-world applications, including:

  • Solving equations and finding roots
  • Simplifying expressions and solving systems of equations
  • Modeling real-world phenomena, such as population growth and chemical reactions
  • Factoring polynomials is also used in cryptography and coding theory.

Q: Can you provide some examples of factoring polynomials?

A: Yes, here are some examples of factoring polynomials:

  • 2x2+5x+3=(2x+1)(x+3)2x^2 + 5x + 3 = (2x + 1)(x + 3)

  • x3−2x2−5x+6=(x−3)(x2+x−2)x^3 - 2x^2 - 5x + 6 = (x - 3)(x^2 + x - 2)

  • x2+4x+4=(x+2)2x^2 + 4x + 4 = (x + 2)^2

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By mastering the techniques used to factor polynomials, we can solve equations, find roots, and simplify expressions. In this article, we have answered some frequently asked questions about factoring polynomials and provided some examples of factoring polynomials.

Final Answer

Factoring polynomials is a powerful tool that has many real-world applications. By understanding the different techniques used to factor polynomials, we can solve equations, find roots, and simplify expressions.