Problem 76 - College Algebra

Question

In Exercises \(57-84\), factor completely, or state that the polynomial is prime. $$48 y^{4}-3 y^{2}$$

Step-by-Step Solution

Answer

The factorised form of \(48y^{4}-3y^{2}\) is \(3y^{2}(4y-1)(4y+1)\).

Step 1: Identify the Greatest Common Factor (GCF)

The first step in factoring is to identify the greatest common factor. In this case, the GCF for \(48y^{4}-3y^{2}\) is \(3y^{2}\).

Step 2: Divide each term by the GCF

The next step is to divide each term of the equation by the GCF: \(\frac{48y^{4}}{3y^{2}} - \frac{3y^{2}}{3y^{2}}\). This simplifies to \(16y^{2} - 1\).

Step 3: Factor the Difference of Squares

The expression \(16y^{2}-1\) is a difference of squares, which can be factored into \((4y-1)(4y+1)\).

Step 4: Final Factor Form

Finally, we multiply the factored form by the GCF we originally took out, leading us to the completely factored form of \(3y^{2}(4y-1)(4y+1)\).

Key Concepts

Greatest Common Factor (GCF)

When dealing with polynomials, one of the first steps to simplify them is to find the Greatest Common Factor, often abbreviated as GCF. The GCF is the largest factor that divides each term of the polynomial without leaving a remainder.
To identify the GCF in a polynomial like \(48y^4 - 3y^2\):

• Look at both the numerical and variable components.
• For the numerical part, find the largest number that divides both coefficients (48 and 3). This is 3.
• For the variable part, determine the smallest power of the common variable (in this case, \(y^2\)).

Thus, the GCF for \(48y^4 - 3y^2\) is \(3y^2\), which you can then factor out from the expression.
This makes it easier to work with and further simplifies the expression.

Difference of Squares

The next step in factoring the polynomial \(48y^4 - 3y^2\) involves recognizing a specific pattern called the Difference of Squares. A difference of squares is a special factoring technique which applies when you have two perfect squares separated by a subtraction sign.
The general formula for the difference of squares is: \[a^2 - b^2 = (a - b)(a + b)\] Here, the expression \(16y^2 - 1\) fits this formula, as \(16y^2\) is \((4y)^2\) and 1 is \(1^2\). This allows us to rewrite the expression as \((4y - 1)(4y + 1)\).
Recognizing and utilizing patterns like the difference of squares helps in breaking down complex expressions into simpler parts.

Factoring Techniques

Factoring polynomials can seem challenging, but there are techniques that make the process more manageable. Let's walk through some helpful strategies when approaching problems like the one described here.

• Start with the GCF: Always begin by factoring out the Greatest Common Factor. This not only simplifies the polynomial but can reveal underlying structures like a difference of squares.
• Look for familiar patterns: Recognizing patterns such as squares, cubes, or other common factorizations (like trinomial squares) can simplify your work.
• Break it down: After factoring out the GCF and recognizing any patterns, further simplify as needed.

The final product ensures that we account for all factors, including any GCF removed initially. After you have factored the polynomial completely, simply multiply back the GCF to get to the final expression: \(3y^2(4y - 1)(4y + 1)\).
Employing these straightforward techniques will enable you to tackle a wide range of polynomial expressions efficiently.