Problem 79 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$2 y^{5}-2 y^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(2 y^{5}-2 y^{2}\) is \(2y^{2}(y - 1)(y^{2} + y + 1)\)

Step 1: Identify the Greatest Common Factor (GCF)

In the polynomial \(2 y^{5}-2 y^{2}\), both terms have a common factor of \(2y^{2}\). This is the greatest common factor.

Step 2: Factor out the GCF

Next, divide the polynomial by the GCF \((2y^{2})\). Doing this, the original polynomial \((2 y^{5}-2 y^{2})\) becomes \(2y^{2}(y^{3} - 1)\).

Step 3: Continue Factoring

The factor \((y^{3} - 1)\) is a difference of cubes, which can be factored further using the formula \(a^{3} - b^{3} = (a-b)(a^{2} + ab + b^{2})\), where \(a = y\) and \(b = 1\). Factoring \((y^{3} - 1)\) results in \((y - 1)(y^{2} + y + 1)\).

Step 4: Write the Completed Factorization

Putting it all together, the completely factored form of the original polynomial is \(2y^{2}(y - 1)(y^{2} + y + 1)\).

Step 5: Verification

The factored form can be verified by expanding it back into the original polynomial through multiplication, or by comparing the graphs of the original polynomial and the factored expression. They should be identical.

Key Concepts

Greatest Common Factor (GCF)

When factoring polynomials, one of the first steps taken is identifying the Greatest Common Factor (GCF), which is the largest expression that divides each term of the polynomial. Think of it as the numerical and variable components that all terms of the polynomial share.

For instance, let’s consider the polynomial in the exercise, 2y^5 - 2y^2. Both terms have a '2' and at least 'y^2' in common. Hence, the GCF is 2y^2. Factoring out the GCF simplifies the polynomial into a more manageable form and is crucial for spotting further factorization opportunities. It's like taking the biggest slice out of a pie and then seeing what specific flavors each smaller piece has.

Difference of Cubes

A difference of cubes occurs when a polynomial is composed of two terms that are both perfect cubes and subtracted from one another. This special form can be factored using the formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

After factoring out the GCF in the given polynomial, we're left with \(y^3 - 1\), which is a difference of cubes; \(y^3\) being \(a^3\) and 1 as \(b^3\). Applying our formula gives us \((y - 1)(y^2 + y + 1)\), which reveals the inner structure of the cubic term. It's like breaking down a building into its individual blocks to understand its architecture better.

Polynomial Factorization Techniques

Factoring polynomials is akin to breaking down a complex puzzle into smaller, more manageable pieces, and various techniques exist depending on the type of polynomial in question. Common methods include finding the GCF, using the difference of squares or cubes formulas, factoring by grouping, and applying the quadratic formula.

What's essential is to be methodical: first look for a GCF, then observe the structure of what remains. Can it be a square or a cube? Perhaps grouping terms exposes a common binomial. Some polynomials can even be factored multiple times using different techniques. Always be on the lookout for the next piece of the puzzle to simplify.

Verification of Factorization

Verifying your factorization makes sure that your 'puzzle pieces' fit perfectly to rebuild the original 'puzzle'. For polynomials, this means expanding the factors and comparing the result to the original polynomial. If they match, the factorization is correct.

In the example, after factorizing the polynomial, we get \(2y^2(y - 1)(y^2 + y + 1)\). To verify, multiply these factors. If the product equals the original \(2y^5 - 2y^2\), the factorization is correct. Alternatively, graphing the original polynomial and the factored expression on a graphing utility and observing if the graphs coincide provides a visual confirmation of correct factorization.