Problem 36 - Beginning and Intermediate Algebra

Question

Factor completely by first taking out \(-1\) and then by factoring the trinomial, if possible. Check your answer. $$-y^{2}-9 y-18$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial is \(-1(y+6)(y+3)\).

Step 1: 1. Factor out -1 from the polynomial

First, we will factor out -1 from the given polynomial: $$-1(y^{2}+9y+18)$$

Step 2: 2. Factor the trinomial

Now we will factor the trinomial inside the parenthesis, \(y^{2}+9y+18\). To do this, we need to look for two numbers that multiply to 18 and add up to 9. Those numbers are 6 and 3. Therefore, the trinomial can be factored as: $$(y+6)(y+3)$$

Step 3: 3. Combine the factors

Now, combine the -1 factor with the factored trinomial to obtain the completely factored polynomial: $$-1(y+6)(y+3)$$

Step 4: 4. Check the answer

To check our answer, we will multiply the factors back together and see if we obtain the original polynomial. First, let's multiply the factored trinomial together: $$(y+6)(y+3) = y^2 + 3y + 6y + 18 = y^2 + 9y + 18$$ Now, multiply by -1: $$-1(y^2 + 9y + 18) = -y^2 - 9y - 18$$ The product matches the original polynomial, so our factorization is correct. The completely factored form of the given polynomial is: $$-1(y+6)(y+3)$$

Key Concepts

Trinomial Factoring

Trinomial factoring is a technique used to break down expressions made up of three terms into simpler multiplicative components. This process helps in simplifying and solving algebraic expressions more efficiently. In the example of the polynomial \(-y^2 - 9y - 18\), the polynomial first needed to be factored by taking \(-1\) out. This was the first step in making the trinomial easier to handle, turning it into

• \(y^2 + 9y + 18\)

Now, the task is to identify two numbers that will multiply to give the constant term (18) and add up to the middle coefficient (9). The correct pair in this case is 3 and 6. Thus, the trinomial can be factored as

• \((y + 6)(y + 3)\)

This factorization allows us to see which simpler terms, when multiplied together, will give the original polynomial inside the parentheses.

Checking Factorization

Just like a double-check in mathematics, checking factorization means verifying that the factors you have found indeed work to create the original expression. Once the polynomial \(-y^2 - 9y - 18\) was rewritten using factors, it is crucial to ensure accuracy.
To do this, the factors \((y+6)(y+3)\) are multiplied back together:

• First, expand \((y + 6)(y + 3)\) to get \(y^2 + 3y + 6y + 18 = y^2 + 9y + 18\).
• Next, multiply this result by \(-1\) to retrieve \(-y^2 - 9y - 18\).

If this operation returns to the original polynomial, then the factorization is confirmed correct. This step ensures there were no errors in the factorization and that the negative sign extracted initially has been correctly managed.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics, consisting of variables, numbers, and operations that together represent a quantity or formula. The polynomial \(-y^2 - 9y - 18\) is an example of a quadratic algebraic expression. Understanding such expressions involves recognizing terms and operations that allow for simplification processes like factoring.
These expressions can appear complex, but they often resolve into more manageable parts through methods like grouping or using specific identities and rules.
For example, in factoring, identifying a negative sign across the entire expression helps isolate or "clean up" terms so that the task becomes finding pairs of numbers that relate in specific ways, such as multiplying to one number and adding to another in a trinomial setup. Breaking down expressions into factors follows the essence of organizing terms to understand different mathematical properties and solving them efficiently.