Problem 72 - Introductory Algebra for College Students

Question

Factor completely. $$10 y^{3}+12 y^{2}+2 y$$

Step-by-Step Solution

Answer

The complete factorization of the polynomial \(10 y^{3}+12 y^{2}+2 y\) is \(2 y(5 y^{2}+ 6 y + 1)\).

Step 1: Identify Common Factor

Looking at the terms in the polynomial \(10 y^{3}\), \(12 y^{2}\), \(2 y\), the common factor is \(2 y\).

Step 2: Factor Out Common Factor

Now, factor out this common factor from each term of the original polynomial, resulting in an equivalent expression: \(2 y(5 y^{2}+ 6 y + 1)\).

Step 3: Identify Further Factoring

Looking at the polynomial \(5 y^{2}+ 6 y + 1\), it is clear it cannot be further factored because it is not a perfect square trinomial (the coefficient of \(y^2\) (5) or the constant term (1) is not a perfect square) and also it does not have rational roots (by the rational root theorem).

Key Concepts

Common Factor

Factoring polynomials often begins with identifying the greatest common factor (GCF) among all terms. The GCF is the largest expression that divides into each term without leaving a remainder. For example, consider the polynomial 10 y^{3} + 12 y^{2} + 2 y. Each term contains a factor of y, and a numerical factor that is a multiple of 2. Thus, the GCF is 2y.

To factor out the common factor, we divide each term by 2y, simplifying the polynomial to 2y(5y^2 + 6y + 1). This step is critical as it simplifies the polynomial and may reveal further factoring opportunities. Additionally, factoring out the common factor is crucial for solving equations, finding zeros of polynomials, and simplifying algebraic expressions.

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic trinomial where the first and last terms are perfect squares and the middle term is twice the product of the square roots of those perfect squares. Formally, it can be written as a^2 + 2ab + b^2, which factors to (a + b)^2. For instance, x^2 + 6x + 9 is a perfect square trinomial because it can be factored into (x + 3)^2.

However, the trinomial 5y^2 + 6y + 1 in the given exercise is not a perfect square trinomial. Neither 5 (coefficient of y^2) nor 1 (constant term) are perfect squares, and the middle term is not twice the product of the square roots of 5 and 1. Therefore, this trinomial does not factor into a simple binomial square but must be examined through other factoring methods or identified as prime if no such methods apply.

Rational Root Theorem

The Rational Root Theorem provides a way to list all possible rational zeros for a polynomial equation with integer coefficients. According to this theorem, any possible rational zero p/q (in the reduced form where p and q are integers and have no common factors other than 1) of a polynomial equation must satisfy the condition where p is a factor of the constant term (the term without a variable) and q is a factor of the leading coefficient (the coefficient of the highest degree term).

For the trinomial 5y^2 + 6y + 1, since it's not a perfect square and cannot be factored more easily, the Rational Root Theorem may help in identifying whether this trinomial has rational solutions. In our case, p would have to be a factor of 1 (±1), and q would have to be a factor of 5 (±1, ±5). By testing these possible rational roots and finding no zeros, we can conclude that this trinomial does not have rational roots, thus making it irreducible over the rational numbers.