Problem 21 - College Algebra

Question

Factor each trinomial, or state that the trinomial is prime. $$ x^{2}-8 x+15 $$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^2 - 8x + 15\) is \((x-3)(x-5)\).

Step 1: Identify the coefficients

In the trinomial, \(x^2 - 8x + 15\), the coefficients of \(x^2\), \(x\), and the constant term are 1, -8 and 15 respectively. That is, \(a = 1\), \(b = -8\), and \(c = 15\).

Step 2: Find two numbers

We need to find two numbers that multiply to give the product of \(a\) and \(c\) (that is, \(1*15 = 15\)) and add up to give \(b\) (that is, -8). The two numbers that satisfy these conditions are -3 and -5.

Step 3: Write the trinomial as a product of binomials

We therefore write the trinomial as the product of two binomials using the two numbers we found: \(x^2 -8x + 15 = (x-3)(x-5)\).

Key Concepts

Polynomial Factoring

Factoring polynomials is one of the fundamental skills in algebra that allows us to break down complex expressions into simpler, more manageable parts. Specifically, when we talk about factoring trinomials, we're usually dealing with a three-term polynomial of the form \( ax^2 + bx + c \). The goal of factoring is to express the trinomial as a product of two binomials, like \( (dx + e)(fx + g) \), where the values of \( d, e, f, \) and \( g \) are to be determined.

A common method for factoring trinomials, especially when the leading coefficient \( a \) is 1, is to look for two numbers that multiply to the constant term \( c \), and add up to the linear coefficient \( b \). This approach hinges on the relationship between the coefficients and the factors. Let's consider an example:

• Given the trinomial \( x^2 - 8x + 15 \), we look for two numbers that multiply to 15 (the constant term) and add to -8 (the coefficient of the linear term).
• We find that the numbers -3 and -5 satisfy this condition, because \( (-3) \times (-5) = 15 \) and \( (-3) + (-5) = -8 \).
• Using these numbers, we can then factor the trinomial as \((x - 3)(x - 5)\).

This process turns a quadratic expression into a form that can easily reveal its roots or solutions when set equal to zero and is crucial for solving quadratic equations.

Quadratic Equations

Quadratic equations are a type of polynomial equation that specifically resemble the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The solutions to these equations, also known as the roots, can be found using various techniques, such as factoring, completing the square, the quadratic formula, or graphing.

When we factor a quadratic equation like the trinomial in our exercise, we effectively find the values of \( x \) that make the equation equal to zero. For the factored form of the trinomial \( x^2 - 8x + 15 \), which is \( (x - 3)(x - 5) \), setting each binomial equal to zero gives the roots of the quadratic equation:

• \( x - 3 = 0 \) leads to \( x = 3 \)
• \( x - 5 = 0 \) leads to \( x = 5 \)

These are the x-intercepts of the quadratic's graph, representing the points where the parabola crosses the x-axis. Understanding quadratic equations is vital for a range of applications in mathematics and science, and factoring is often the quickest method for solving them when possible.

Coefficients in Algebra

In algebra, coefficients are the numerical factors attached to terms within an algebraic expression. They provide fundamental information about the term's contribution to the overall expression. For example, in the trinomial \( ax^2 + bx + c \), the coefficient \( a \) is connected to the squared term, \( b \) to the linear term, and \( c \) is the constant term without a variable.

In factoring trinomials, these coefficients play a pivotal role in determining the numbers we select for the factoring process. Specifically, in the exercise \( x^2 - 8x + 15 \), we identify that the coefficient of \( x^2 \) is 1 (often this is implied and not explicitly written), the coefficient of \( x \) is -8, and the constant term is 15. These values lead us to seek out the pair of numbers that allow us to split the middle term - in this case, -8x - into two parts for factoring.

A strong comprehension of how coefficients influence algebraic expressions is essential for delving into more complex mathematical topics. Coefficients dictate the shape and position of a graph of an equation, and they are instrumental when applying algebra to solve real-world problems. Whenever we approach a problem involving polynomial factoring or solving equations, a close examination of the coefficients sets the foundation for our subsequent steps in the problem-solving process.