Problem 55 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$-4 y^{3}+28 y^{2}-40 y$$

Step-by-Step Solution

Answer

The fully factorized form of the polynomial \(-4 y^{3}+28 y^{2}-40 y\) is \(-4y(y-2)(y-5)\).\n

Step 1: Arrange the Polynomial

The Polynomial given is \(-4 y^{3}+28 y^{2}-40 y\). Good rule of thumb with dealing these types of exercises is arranging the terms in descending order of their degrees.

Step 2: Find the Greatest Common Factor

The GCF is the highest number that divides exactly into two or more numbers. In this case, the GCF of the coefficients of the given polynomial is 4 and the smallest power of y (since all terms involve y) is y. Hence, the GCF between these three terms is \(4y\).

Step 3: Divide by the GCF

The next step is to divide the entire polynomial by the GCF, which is \(4y\). This yields the expression \(-y^{2}+7y-10\).

Step 4: Factorize the Quadratic Polynomial

The quadratic (-y²+7y-10) can be factored by finding two numbers that multiply together to give -10 (the last term) and at the same time add up to 7 (the coefficient of the middle term). The two numbers are -2 and 5. Therefore, \(-y^{2}+7y-10\) can be written as \(-(y-2)\(y-5)\)

Step 5: Write the Final Answer

Reintroducing the GCF we divided out in step 3, the completely factored form of the given polynomial \(-4 y^{3}+28 y^{2}-40 y\) is \(-4y(y-2)(y-5)\). This is the final factorized form of the given polynomial.

Key Concepts

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is a crucial step in simplifying expressions, especially when dealing with polynomials. It refers to the highest number that divides exactly into two or more numbers without leaving a remainder. In a polynomial context, it includes both numerical coefficients and variables.

In the given polynomial expression \[-4 y^{3}+28 y^{2}-40 y\], the goal is to find the largest expression that can be factored out of each term. Here's how to find it:

• Identify the coefficients of each term. In our polynomial, these are -4, 28, and -40.
• Determine the greatest number that divides all coefficients. Here, 4 fits the bill as the GCF.
• Check the variable in common across all terms. Since each term includes at least one 'y', 'y' forms part of the GCF.
• Resulting GCF is thus \(4y\).

This GCF allows us to "simplify" the polynomial by dividing each term by \(4y\), preparing it for further factorization steps.

Quadratic Polynomial

A quadratic polynomial is an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. It represents a polynomial of degree 2 and the term 'quadratic' signifies that the highest power of the variable is 2.

In the problem, once we divided the polynomial by the GCF, we got a quadratic expression:\(-y^2 + 7y - 10\). To factor this, we look for two numbers that fit two conditions:

• They multiply together to give the last term (here, -10).
• They add up to the middle term (here, 7).

After checking combinations, we discover that -2 and 5 satisfy both conditions, giving us the factorization:\(-(y - 2)(y - 5)\).Recognizing and manipulating quadratic polynomials is a key skill in algebra that aids in solving equations and analyzing graphs.

Factored Form

When you write a polynomial as a product of its factors, you express it in its factored form. This is beneficial because it reveals the roots of the polynomial and greatly simplifies various algebraic operations. In essence, factoring breaks down complex expressions into simpler, multiplicative components.

For the polynomial \(-4 y^{3}+28 y^{2}-40 y\), the journey to the factored form involved several steps:

• Extract the Greatest Common Factor \(4y\).
• Factor the resulting quadratic expression \(-(y - 2)(y - 5)\).

By recombining these factors with the GCF, the complete factored form is:\(-4y(y - 2)(y - 5)\). This expression reveals that the polynomial equals zero when \(y\) is 0, 2, or 5. This notation is handy for graphing or solving equations, aiding both in theoretical and practical applications.