Problem 76 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ -9 y^{4}-3 y^{3}+6 y^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(3y^{2}(-3y^{2} - y + 2)\), where the trinomial is prime.

Step 1: Identify a Greatest Common Factor (GCF)

The first step in factoring is to determine the greatest common factor of all terms. In the given expression \(-9y^{4} - 3y^{3} + 6y^{2}\), each term has a common factor of \(3y^{2}\). We factor out \(3y^{2}\) from each term to simplify the expression.

Step 2: Factor Out the GCF

Divide each term in the expression by the GCF \(3y^{2}\):\[-9y^{4} - 3y^{3} + 6y^{2} = 3y^{2}(-3y^{2}) + 3y^{2}(-y) + 3y^{2}(2)\].Thus, we can factor the expression as:\[3y^{2}(-3y^{2} - y + 2)\].

Step 3: Check for Further Factorization

Now, examine the expression in parentheses \(-3y^{2} - y + 2\) to see if it can be factored further, for example, as a trinomial. In this case, no two numbers multiply to \(-6\) (the product of \(-3\times 2\)) and add to \(-1\) (the middle term). Therefore, \(-3y^{2} - y + 2\) is prime.

Step 4: Write the Final Factored Expression

Since the expression inside the parentheses cannot be factored further, the fully factored form of the original expression is:\[3y^{2}(-3y^{2} - y + 2)\].

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor, or GCF, is fundamental in helping simplify polynomial expressions. The GCF of a set of terms is the largest factor that is shared by all of the terms involved. Identifying it is the first step when factoring any expression. This process helps in extending basic arithmetic skills to more complex algebraic expressions.

To find the GCF of a polynomial, you need to look at both the numerical coefficients and the variables. For example, in \(-9y^4 - 3y^3 + 6y^2\), the numerical part includes the numbers 9, 3, and 6, which all share a factor of 3. Similarly, each term has a noticeable variable factor of \(y^2\). Thus, the GCF here is \(3y^2\).

Finding and factoring out the GCF simplifies the expression, making it easier to work with. It reduces complex equations into simpler ones that are often more manageable and insightful for solving other algebraic tasks.

Prime Expression

A prime expression in algebra refers to a polynomial that cannot be factored further using integer coefficients. This concept parallels the idea of prime numbers, which are only divisible by 1 and themselves. Identifying whether an expression is prime is a key step when factoring.

When examining the expression \(-3y^2 - y + 2\), once the GCF has been factored out, you check for further factorization. You attempt to find two numbers that, when multiplied, yield the product of the first and last coefficients, and when added, give the middle coefficient.

In our case here: there are no such numbers for \(-6\) and \(-1\). This informs us that the trinomial is prime because it cannot be broken down into simpler polynomial factors. This understanding is essential, as it tells you when you've reached the simplest form of your expression within integer coefficients.

Factorization Techniques

Factorization is a powerful technique in algebra that involves breaking down expressions into simpler, multivariable components or factors. The process streamlines finding solutions and understanding the properties of algebraic expressions.

The methods for successful factorization include:

• Factoring out the GCF, which simplifies all terms at once.
• Decomposing quadratics, using methods like "splitting the middle term" or finding roots through factorization.
• Recognizing special patterns, such as difference of squares or perfect trinomials.

In the provided solution, after factoring out the GCF, further factorization techniques were applied to examine the trinomial \(-3y^2 - y + 2\). The attempt to break it down was crucial to confirm it as a prime expression.

Learning these techniques helps make better sense of mathematical structures and strengthens overall problem-solving abilities in algebra.