Completely Factor The Polynomial -4y³ + 8y² - 6y² + 12y. How To Factor -4y³ + 8y² - 6y² + 12y Completely?

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-4y³ + 8y² - 6y² + 12y

Factoring polynomials is a fundamental skill in algebra. It involves breaking down a polynomial expression into a product of simpler expressions, which can be incredibly useful for solving equations, simplifying expressions, and understanding the behavior of functions. In this comprehensive guide, we will walk through the process of completely factoring the polynomial -4y³ + 8y² - 6y² + 12y. This example will illustrate key techniques like identifying the greatest common factor (GCF) and factoring by grouping. Let's dive in and master the art of polynomial factorization.

Understanding the Importance of Factoring Polynomials

Before we delve into the specifics of factoring the given polynomial, it's crucial to understand why this skill is so important in mathematics. Factoring polynomials is not just an abstract algebraic exercise; it has practical applications in various areas of mathematics and beyond. Here are some key reasons why mastering polynomial factorization is essential:

  • Solving Equations: Factoring is often the first step in solving polynomial equations. By breaking down a polynomial into its factors, we can set each factor equal to zero and find the roots or solutions of the equation. This is particularly useful for quadratic equations and higher-degree polynomials.
  • Simplifying Expressions: Factoring can simplify complex algebraic expressions. By factoring out common factors or using factoring patterns, we can reduce the expression to a more manageable form, making it easier to work with.
  • Graphing Functions: The factored form of a polynomial can provide valuable information about the graph of the corresponding function. The roots of the polynomial (where the function equals zero) are easily identified from the factored form, and these roots correspond to the x-intercepts of the graph.
  • Calculus Applications: In calculus, factoring polynomials is essential for finding limits, derivatives, and integrals of polynomial functions. It also plays a role in solving optimization problems and analyzing the behavior of functions.
  • Real-World Applications: Polynomials and their factored forms appear in various real-world applications, such as modeling physical phenomena, designing engineering systems, and analyzing financial data. Factoring skills can be applied to solve problems in these areas.

In essence, factoring polynomials is a gateway to understanding and manipulating algebraic expressions and equations. It is a skill that will serve you well throughout your mathematical journey.

Step 1: Simplify the Polynomial

Before we begin the factoring process, it's always a good practice to simplify the polynomial by combining like terms. This makes the subsequent steps easier and reduces the risk of errors. In our case, the polynomial is:

-4y³ + 8y² - 6y² + 12y

We can see that there are two terms with y²: 8y² and -6y². Let's combine these terms:

-4y³ + (8y² - 6y²) + 12y

-4y³ + 2y² + 12y

Now, we have a simplified polynomial: -4y³ + 2y² + 12y. This expression is equivalent to the original, but it's in a more manageable form.

Step 2: Identify and Factor Out the Greatest Common Factor (GCF)

The next crucial step in factoring any polynomial is to identify and factor out the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all the terms of the polynomial. This step simplifies the polynomial and makes further factoring easier. To find the GCF, we need to consider both the coefficients (the numerical parts of the terms) and the variables.

In our simplified polynomial, -4y³ + 2y² + 12y, let's analyze the coefficients: -4, 2, and 12. The greatest common factor of these numbers is 2. However, since the leading coefficient is negative (-4), it is generally preferred to factor out a negative GCF. Thus, we'll factor out -2.

Now, let's look at the variable parts: y³, y², and y. The lowest power of y that appears in all terms is y (which is y¹). Therefore, the greatest common variable factor is y.

Combining the numerical and variable factors, the GCF of the polynomial -4y³ + 2y² + 12y is -2y. Now, we factor out -2y from each term:

-2y(2y² - y - 6)

Notice how we divided each term of the original polynomial by -2y and wrote the result inside the parentheses. This step is crucial for preserving the equivalence of the expression.

Step 3: Factor the Remaining Quadratic Expression

After factoring out the GCF, we are left with the expression 2y² - y - 6 inside the parentheses. This is a quadratic expression, which is a polynomial of degree 2. To factor a quadratic expression of the form ax² + bx + c, we need to find two numbers that multiply to ac and add up to b. In our case, a = 2, b = -1, and c = -6.

So, we need two numbers that multiply to (2)(-6) = -12 and add up to -1. The numbers -4 and 3 satisfy these conditions (-4 * 3 = -12 and -4 + 3 = -1). Now, we rewrite the middle term (-y) using these numbers:

2y² - 4y + 3y - 6

Next, we factor by grouping. We group the first two terms and the last two terms:

(2y² - 4y) + (3y - 6)

Now, factor out the GCF from each group:

2y(y - 2) + 3(y - 2)

Notice that we now have a common factor of (y - 2) in both terms. We factor this out:

(y - 2)(2y + 3)

So, the factored form of the quadratic expression 2y² - y - 6 is (y - 2)(2y + 3).

Step 4: Write the Completely Factored Polynomial

Now that we have factored the quadratic expression, we can write the completely factored form of the original polynomial. Remember that we factored out -2y in Step 2. So, the completely factored form is:

-2y(y - 2)(2y + 3)

This is the final answer. We have successfully factored the polynomial -4y³ + 8y² - 6y² + 12y completely.

Summary of the Factoring Process

Let's recap the steps we took to completely factor the polynomial -4y³ + 8y² - 6y² + 12y:

  1. Simplify the Polynomial: Combine like terms to simplify the expression. In our case, we simplified -4y³ + 8y² - 6y² + 12y to -4y³ + 2y² + 12y.
  2. Identify and Factor Out the Greatest Common Factor (GCF): Find the largest factor that divides evenly into all terms and factor it out. We factored out -2y from -4y³ + 2y² + 12y to get -2y(2y² - y - 6).
  3. Factor the Remaining Quadratic Expression: Factor the quadratic expression inside the parentheses. We factored 2y² - y - 6 into (y - 2)(2y + 3).
  4. Write the Completely Factored Polynomial: Combine the GCF and the factored quadratic expression to get the final factored form. The completely factored form of -4y³ + 8y² - 6y² + 12y is -2y(y - 2)(2y + 3).

Common Mistakes to Avoid When Factoring Polynomials

Factoring polynomials can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to watch out for:

  • Forgetting to Factor Out the GCF: Always start by looking for the greatest common factor. If you skip this step, you may end up with a more complicated expression to factor later.
  • Incorrectly Identifying the GCF: Make sure you find the greatest common factor. If you factor out a smaller factor, you'll still need to factor the remaining expression further.
  • Sign Errors: Be careful with signs when factoring out negative factors or when using factoring patterns. A simple sign error can lead to an incorrect answer.
  • Factoring Too Quickly: Don't rush through the process. Take your time to analyze the polynomial and choose the appropriate factoring techniques.
  • Not Checking Your Answer: After factoring, you can check your answer by multiplying the factors back together. If you get the original polynomial, you've factored correctly.

By being aware of these common mistakes, you can improve your factoring skills and avoid errors.

Conclusion: Mastering Polynomial Factoring

Factoring polynomials is a fundamental skill in algebra that opens the door to solving equations, simplifying expressions, and understanding the behavior of functions. In this guide, we've walked through the process of completely factoring the polynomial -4y³ + 8y² - 6y² + 12y, illustrating key techniques such as identifying the greatest common factor (GCF) and factoring quadratic expressions. Remember to always start by simplifying the polynomial and looking for the GCF. When factoring quadratic expressions, use the appropriate factoring patterns or techniques. By practicing these steps and avoiding common mistakes, you can master polynomial factoring and confidently tackle more complex algebraic problems. Keep practicing, and you'll find that factoring polynomials becomes second nature.