Problem 63 - Introductory Algebra for College Students

Question

Factor completely. $$4 y^{2}+2 y-30$$

Step-by-Step Solution

Answer

The completely factored form of \(4y^2+2y-30\) is \(2(y+5)(y-3)\).

Step 1: Extract the Common Factor

The given expression is \(4y^2+2y-30\). The common factor for these three terms is 2. Extracting the common factor gives us \(2(2y^2 + y -15)\).

Step 2: Factor the Trinomial

Factor the trinomial \(2y^2 + y - 15\). Looking for two numbers that multiply to -30 (product of 2 and -15) and add to 1 (the coefficient of y). These numbers are 5 and -6. So the factored form is \(2(y + 5)(y - 3)\)

Key Concepts

Extracting Common Factors

When faced with a polynomial such as \(4y^2+2y-30\), the first step in simplification is to look for a common factor. A common factor is a number or expression that is a divisor of all the terms of the polynomial. In our example, the number 2 is a factor of each term. Extracting this common factor simplifies the polynomial to \(2(2y^2 + y - 15)\).

This step is crucial because it reduces the complexity of the polynomial, making further steps in the factoring process easier. Always check for a common factor before attempting other factoring methods, as it can save time and reveal simpler patterns in the polynomial.

Factoring Quadratic Equations

After extracting the common factor, the next step is to factor the resulting quadratic equation. Factoring a quadratic equation involves breaking it down into a product of two binomials. For the trinomial \(2y^2 + y - 15\), we need to find two numbers whose product is equal to the coefficient of \(y^2\) times the constant term (in this case, \(2 \times -15 = -30\)) and whose sum equals the coefficient of the \(y\) term (in this case, 1).

These two numbers are 5 and -6. When placing these numbers in binomial factors, we get \(2(y + 5)(y - 3)\). It's like solving a puzzle: the correct pieces will fit together so that when multiplied out, they recreate the original quadratic equation.

Polynomial Factorization

Polynomial factorization refers to the process of breaking down a complex polynomial into simpler, easily solvable expressions. When we factor a polynomial, we express it as the product of two or more polynomials. The factored form of \(4y^2+2y-30\), using the steps aforementioned, is \(2(y + 5)(y - 3)\).

This factored form is very useful because it reveals the roots or zeros of the polynomial, which can be critical in solving equations and graphing the function. Factoring is not just about finding a solution; it's about understanding the structure of the polynomial. By factoring completely, we gain deeper insight into the behavior of the polynomial and how it can be manipulated for different mathematical applications.