Problem 8 - Algebra A Combined Function

Question

Factor each polynomial by grouping. Notice that Step 3 has already been done in these exercises. $$ 2 y^{4}-10 y^{2}+7 y^{2}-35 $$

Step-by-Step Solution

Answer

The polynomial \(2y^4 - 10y^2 + 7y^2 - 35\) factors to \((2y^2 + 7)(y^2 - 5)\).

Step 1: Group the terms

Divide the polynomial into two groups. The polynomial given is \(2y^4 - 10y^2 + 7y^2 - 35\). We can group it as \((2y^4 - 10y^2) + (7y^2 - 35)\).

Step 2: Factor out the greatest common factor in each group

For the first group \((2y^4 - 10y^2)\), the greatest common factor is \(2y^2\), so it becomes \(2y^2(y^2 - 5)\). For the second group \((7y^2 - 35)\), the greatest common factor is \(7\), so it becomes \(7(y^2 - 5)\).

Step 3: Factor by grouping

Now, since both terms contain the common factor \((y^2 - 5)\), the expression \(2y^2(y^2 - 5) + 7(y^2 - 5)\) can be factored as \((2y^2 + 7)(y^2 - 5)\).

Key Concepts

Grouping Method

The grouping method is a powerful strategy for factoring polynomials, especially when other methods may be less effective. This technique involves organizing the terms in the polynomial into smaller groups, which can then be factored individually. Let’s dive into how this method works using our example polynomial:Given the polynomial: \[ 2y^4 - 10y^2 + 7y^2 - 35 \]The first step in the grouping method is to divide this polynomial into two manageable groups:\[ (2y^4 - 10y^2) + (7y^2 - 35) \] Why group this way? The rationale is to find smaller sets of terms that might share common factors. Grouping creates a structure that makes it easier to see factorization opportunities. Within each of these grouped terms, you can work towards breaking it down further, ultimately simplifying the entire polynomial.

Greatest Common Factor

The Greatest Common Factor (GCF) plays a key role in polynomial factorization. It is the largest factor shared by all coefficients and variables within a set of terms. Finding the GCF helps simplify each group in the polynomial. Applying this to our grouped expression: In the first group: \[ (2y^4 - 10y^2) \] The terms have a GCF of \(2y^2\). Factoring this out gives: \[ 2y^2(y^2 - 5) \] For the second group: \[ (7y^2 - 35) \] The GCF here is \(7\). Factoring it out yields: \[ 7(y^2 - 5) \] Locating and extracting the GCF is crucial for simplifying each group, which reveals a common factor that facilitates further factorization.

Factoring Techniques

Factoring techniques are strategies employed to rewrite polynomials as products of simpler expressions. Once we have applied the grouping method and extracted the greatest common factor, like in our example, we use a common factor to combine the expression.After grouping and factoring out the GCF, we have:- \(2y^2(y^2 - 5) + 7(y^2 - 5)\)Notice both terms contain the common factor \((y^2 - 5)\). This shared factor allows us to simplify by further combining into a factored form:- \((2y^2 + 7)(y^2 - 5)\)This technique not only simplifies the expression but also expresses the polynomial in a product form, making it easier to solve equations and understand the behavior of functions.