Problem 70 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Begin by asking yourself, "Can I factor out a GCF?" $$10 y^{5}+50 y^{4}+60 y^{3}$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial $$10y^5 + 50y^4 + 60y^3$$ is \(10y^3(y + 2)(y + 3)\).

Step 1: Identify the GCF of the terms

To find the GCF of the polynomial terms, we can examine the coefficients and the variables separately. For the coefficients 10, 50, and 60, we see that their GCF is 10. For the variables, we see that each term has a factor of \(y^3\). Thus, the GCF for the whole polynomial is 10\(y^3\).

Step 2: Factor out the GCF

Now, let's factor out the GCF from the original polynomial: $$10y^5 + 50y^4 + 60y^3 = 10y^3(y^2 + 5y + 6)$$

Step 3: Factor the quadratic polynomial

Next, we observe the quadratic polynomial inside the parentheses, \(y^2 + 5y + 6\). To factor this, we are looking for two numbers whose product is equal to the constant term (6) and whose sum is equal to the coefficient of the linear term (5). The two numbers meeting these criteria are 2 and 3. Therefore, we can factor the quadratic as \((y + 2)(y + 3)\).

Step 4: Write the complete factored form

Finally, we combine the factored GCF and the factored quadratic polynomial to obtain the completely factored form of the original polynomial: $$10y^5 + 50y^4 + 60y^3 = 10y^3(y^2 + 5y + 6) = 10y^3(y + 2)(y + 3)$$

Key Concepts

Greatest Common Factor (GCF)

Before diving into more complex factoring, we always start by identifying the Greatest Common Factor (GCF) of a polynomial. The GCF is the highest number or expression that can be evenly divided into each term of the polynomial. We split this process into two parts: coefficients and variables. For coefficients like 10, 50, and 60, we identify the number 10 as the GCF. For any shared variables, we take the lowest exponent, in this case, it is \(y^3\). Therefore, the GCF for \(10y^5 + 50y^4 + 60y^3\) is \(10y^3\). By factoring out the GCF, we simplify the polynomial and make subsequent steps much easier. Remember, factoring out the GCF sets the stage for tackling more complicated expressions within the polynomial.

Quadratic Polynomial

Once we've factored out the GCF, we often encounter a quadratic polynomial, like \(y^2 + 5y + 6\) in this problem. A quadratic polynomial is a polynomial of degree 2, typically in the form \(ax^2 + bx + c\). The main goal here is to express this quadratic as a product of two binomials, which involves finding two numbers that multiply to the constant term \(c\) and add up to the coefficient of the linear term \(b\).
For instance, for \(y^2 + 5y + 6\), we look for two numbers whose multiplication gives us 6 and addition results in 5. The numbers 2 and 3 satisfy these conditions because \(2 \times 3 = 6\) and \(2 + 3 = 5\). Thus, \(y^2 + 5y + 6\) can be factored into \((y + 2)(y + 3)\). This technique makes it possible to handle and solve quadratic polynomials through factoring.

Factoring Techniques

Factoring techniques involve a systematic approach to breaking down a polynomial into simpler components called factors. The process starts with recognizing and extracting the GCF, simplifying what’s left. This was apparent in the polynomial \(10y^5 + 50y^4 + 60y^3\), where factoring out \(10y^3\) simplified further steps.
The next step, in most scenarios, is to factor any resulting quadratic. Techniques like the method of finding two numbers that multiply to the constant term and add to the linear coefficient are key. Additionally, understanding special cases like perfect square trinomials and differences of squares further expands your toolkit.

• Always begin with identifying and factoring out the GCF.
• Focus on simplifying resulting expressions into quadratics which often require specific factoring methods.

This structured approach ensures even the most complex polynomials are neatly broken into their simplest factors, facilitating easier solving and comprehension.