Problem 8 - Algebra A Combined Function

Question

Factor each trinomial completely. If a polynomial can't be factored, write "prime." $$ x^{2}-x-30 $$

Step-by-Step Solution

Answer

The trinomial \(x^2 - x - 30\) factors to \((x + 5)(x - 6)\).

Step 1: Understand the Trinomial Structure

The trinomial given is of the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = -1 \), and \( c = -30 \). Our goal is to find two numbers that multiply to \( c = -30 \) and add to \( b = -1 \).

Step 2: Find Two Numbers

We need to find two numbers whose product is \(-30\) and whose sum is \(-1\). After testing factor pairs, we find that \(5\) and \(-6\) multiply to \(-30\) and add to \(-1\).

Step 3: Factor the Trinomial

Using the two numbers found, \(5\) and \(-6\), we can write the trinomial as the product of two binomials: \((x + 5)(x - 6)\).

Step 4: Verify the Factorization

Expand \((x + 5)(x - 6)\) to ensure that it simplifies back to the original trinomial. This expansion gives us \(x^2 - 6x + 5x - 30 = x^2 - x - 30\), confirming that our factorization is correct.

Key Concepts

Trinomial Factorization

Trinomial factorization involves breaking down a quadratic expression into the product of two simpler binomials. Think of it as reverse-engineering an expansion process. This particular exercise starts with a trinomial in the form \(ax^2 + bx + c\). The key is to look for two numbers that multiply to the product of \(a\) and \(c\) and add up to \(b\).

• Identify the coefficients: \(a = 1\), \(b = -1\), and \(c = -30\).
• Find two numbers that multiply to \(a \times c = -30\) and add to \(b = -1\).
• The solution provides \(5\) and \(-6\) as the correct pairs, which allows the expression to be factored as \((x + 5)(x - 6)\).

After identifying these crucial numbers, the process of splitting the middle term and grouping leads to a streamlined factorization. It allows the expression \(x^2 - x - 30\) to be rewritten effectively.

Algebraic Expressions

In algebra, expressions are collections of numbers, variables, and operations. They form the building blocks of many mathematical concepts. An algebraic expression like \(x^2 - x - 30\) isn't just numbers and letters; it represents a particular relation, often in terms of variables and constants.
To understand the factorization of an expression, first interpret its components:

• \(x^2\) is a quadratic term, which is the highest power in a trinomial.
• \(-x\) is the linear term, helping determine how the graph behaves and slopes.
• \(-30\) is the constant that shifts the graph vertically on the coordinate plane.

Understanding these parts can help in visualizing how changes to the expression impact its graph or how it can be manipulated in equations.

Polynomial Equations

Polynomial equations contain polynomial expressions set equal to each other. The term 'polynomial' refers to expressions made up of variables and coefficients combined using operations like addition, subtraction, and multiplication.
For example, the polynomial equation derived from \(x^2 - x - 30 = 0\) can be solved to find the values of \(x\) where the graph intersects the x-axis.

• Once factored, the equation becomes \((x + 5)(x - 6) = 0\). This product equals zero implies zero-product property application.
• Setting each binomial to zero: \(x + 5 = 0\) or \(x - 6 = 0\), provides solutions \(x = -5\) and \(x = 6\).

This equation-solving process highlights the practical use of factorization, enabling determination of solutions, often referred to as 'roots,' that satisfy the original polynomial equation.