Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 3 y^{3}-18 y^{2}-48 y $$

Short answer

3y(y-8)(y+2)

Step-by-step solution

Identify a common factor

Look for the greatest common factor (GCF) of all the terms in the polynomial. For the polynomial 3y^{3}-18y^{2}-48y, we notice that each term is divisible by 3y.

Factor out the common factor

Factor out 3y from each term. This means we divide each term by 3y: 3y(y^{2}-6y-16).

Factor the quadratic expression

Now, factor the quadratic expression y^{2}-6y-16. We need two numbers that multiply to -16 (the constant term) and add to -6 (the coefficient of the linear term). These numbers are -8 and 2. Therefore, we can write the quadratic expression as: (y-8)(y+2).

Combine the factors

Combine the common factor with the factored quadratic expression to get the completely factored form of the polynomial: 3y(y-8)(y+2). This is the fully factored form of the polynomial.

Key concepts

greatest common factor

When factoring polynomials, the first step is to identify the Greatest Common Factor (GCF). The GCF is the largest factor that divides all the terms in the polynomial. Think of it as the greatest number or variable expression that all parts of the polynomial have in common. For example, in the polynomial \(3y^3 - 18y^2 - 48y\), each term is divisible by \(3y\). This means 3y is the GCF. Finding the GCF simplifies the expressions and makes further factoring easier. To find the GCF:

• List the factors of each term.
• Identify the highest factor common to all terms.

quadratic expression

The next step after factoring out the GCF is to factor the remaining expression, often a Quadratic Expression. Quadratic expressions typically take the form \(ax^2 + bx + c\). In our example, after factoring out the GCF, we are left with the quadratic expression \(y^2 - 6y - 16\). To factor this, we need two numbers that multiply to the constant term \(-16\) and add to the coefficient of the linear term \(-6\). In this case, the numbers are \(-8\) and \(2\). Thus, the quadratic expression can be rewritten and factored as \((y - 8)(y + 2)\). Factoring quadratics often involves:

• Identifying the two numbers that meet the multiplication and addition criteria.
• Rewriting the quadratic expression using these numbers.

factored form

After finding the factors, the final step is to write the polynomial in its Factored Form. This means combining all the factors identified in the previous steps. For our polynomial \(3y^3 - 18y^2 - 48y\), we found that the GCF is \(3y\) and the quadratic expression \(y^2 - 6y - 16\) factors into \((y - 8)(y + 2)\). Combining these, the completely factored form is \(3y(y - 8)(y + 2)\). The factored form reveals the roots and solutions of the polynomial, making it easier to solve equations. To summarize, writing a polynomial in factored form involves:

• Factoring out the GCF.
• Factoring any remaining quadratic expressions.
• Combining all the factors into one expression.