Problem 20 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. $$ y^{2}-3 y-18 $$

Step-by-Step Solution

Answer

The factored form of the trinomial \(y^{2}-3 y-18\) is \((y-6)(y+3)\).

Step 1: Identify the coefficients and constant

The given trinomial is \(y^{2}-3 y-18\). Here, the coefficient of \(y^{2}\), \(a = 1\). The coefficient of \(y\), \(b = -3\). And the constant term, \(c = -18\).

Step 2: Find two numbers

We need to find two numbers that multiply together to make \(ac = 1 * -18=-18\), and add together to make \(b = -3\). Those numbers are -6 and 3 since \(-6 * 3 = -18\) and \(-6 + 3 = -3\).

Step 3: Factoring the expression

Rewrite the middle term as the sum of the products of the two numbers found in step 2 and the variable (y). Then factor by grouping. So, \(y^{2}-3 y-18\) becomes \(y^{2}-6y+3y -18\), which factors as \((y-6)(y+3)\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra, and they come in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations represent parabolas when graphed on a coordinate plane.

Key characteristics of quadratic equations include:

• The highest degree is 2, which is what makes it a quadratic.
• They can have zero to two real solutions, depending on the discriminant \(b^2 - 4ac\).
• They can be solved using various methods like factoring, completing the square, or the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

In our exercise, the quadratic equation \(y^2 - 3y - 18 = 0\) is already set in a form that invites factorization, as opposed to using the quadratic formula or completing the square.

Understanding these equations is crucial as they appear frequently in different areas of science, engineering, and even financial models.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations—such as addition and multiplication. They are like phrases in math, representing a value or set of values.

In the exercise, \(y^2 - 3y - 18\) is an algebraic expression. It consists of:

• Variables (\(y\))
• Coefficients (The constant numbers multiplying the variables, like \(-3\) with \(y\))
• Constants ( like \(-18\))
• Operations (Addition and subtraction)

When working with algebraic expressions, understanding how to manipulate them is crucial. Factoring is one key skill, allowing us to simplify expressions or solve equations. This technique helps in breaking down expressions into products of simpler expressions.

For example, in our problem, instead of solving the trinomial equation directly, we rewrite it in a factored form to find the solutions more efficiently.

Polynomial Factorization

Polynomial factorization is a method of breaking down a polynomial into simpler "factor" polynomials, whose product equals the original polynomial. It's like the reverse process of expanding an expression.

In our exercise, the polynomial \(y^2 - 3y - 18\) is factorable into \((y-6)(y+3)\). Here's how it works:

• Identify two numbers that multiply to give the product of the constant coefficient \(ac\) and add to the linear coefficient \(b\).
• Use these numbers to split the middle term \(-3y\) into two separate terms.
• Factor by grouping, which involves grouping the terms to expose a common factor.
• Once groups are made, finding common factors and factoring them out gives you the product of two binomials, or simpler polynomials.

Factorization not only helps solve equations but also simplifies expressions for easier interpretation and manipulation.

This essential skill enhances problem-solving in various algebraic contexts, from quadratic equations to complex calculations involving polynomial identities.