Problem 74 - Introductory Algebra for College Students

Question

Factor completely. $$50 y^{3}+20 y^{2}+2 y$$

Step-by-Step Solution

Answer

The completely factorial form of the polynomial \(50y^3 + 20y^2 + 2y\) is \(2y(25y^2 + 10y + 1)\).

Step 1 Identify Common Factor

Let's first find the common factor of all the terms which turns out to be \(2y\). So, the expression can be written as \(2y(25y^2+10y+1)\).

Step 2 Factorise further

Now, further factorize the expression inside the parenthesis, which is a quadratic equation. But here factorization doesn't give us any further factors, as there are no two numbers that can be multiplied to get 25 and added to get 10.

Step 3 Complete Factorization

So, our completely factorial form of the expression is \(2y(25y^2+10y+1)\).

Key Concepts

Common Factors

When dealing with algebraic expressions, looking for a common factor is a crucial first step. A common factor is a number or variable that divides each term in the expression without leaving a remainder.

• In the expression, \(50y^3 + 20y^2 + 2y\), each term shares the factor \(2y\).
• This means 2 can divide all numerical coefficients (50, 20, 2), and \(y\) is present in each term.

By factoring out \(2y\), we simplify the problem to \(2y(25y^2 + 10y + 1)\), making it easier to work with.
Remember, factoring out common factors not only simplifies expressions but also makes further steps in solving, such as complete factorization, more transparent.

Quadratic Equations

Once the common factor is factored out, sometimes the expression inside the parenthesis forms a quadratic equation. A quadratic equation is a polynomial of degree 2, usually having the standard form \(ax^2 + bx + c\).

• In this exercise, the quadratic equation is \(25y^2 + 10y + 1\).
• Solving or simplifying a quadratic equation often requires finding two numbers that multiply to the product of \(a\) and \(c\), and add up to \(b\).

However, if such numbers don't exist for the given quadratic equation, as is the case here, it indicates that it cannot be simplified by factorization. The equation remains \(25y^2 + 10y + 1\) without further factoring.
This highlights that not all quadratic expressions can be further simplified through traditional factoring.

Complete Factorization

Completing the factorization of an algebraic expression involves ensuring that it is broken down into its component parts fully.

• Initially, this means identifying and extracting common factors as we did with \(2y\).
• Next, checking if the remaining expression, like our quadratic equation \(25y^2 + 10y + 1\), can be further divided into factors.

In our example, after factoring out the common factor, no further factorization of \(25y^2 + 10y + 1\) was possible.
Thus, the complete factorization of the original expression is \(2y(25y^2 + 10y + 1)\).
Understanding complete factorization helps in simplifying expressions fully, which is invaluable for solving mathematical equations efficiently.