Problem 40 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ x^{3}-8 x^{2}+15 x $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial $$x^{3}-8 x^{2}+15 x$$ is $$x(x-3)(x-5)$$.

Step 1: Identify the common term

First, we need to identify the common term in the given trinomial $$x^{3}-8 x^{2}+15 x$$. Here, the common term is $$x$$.

Step 2: Factor out the common term

Now, let's factor out the common term, $$x$$, from the trinomial. $$ x^{3}-8 x^{2}+15 x = x(x^{2}-8x+15) $$

Step 3: Identify the factors of the quadratic expression

Next, we'll find the factors of the quadratic expression inside the parentheses, $$x^2-8x+15$$. We will look for two numbers whose product is equal to the constant term (15) and whose sum is equal to the coefficient of the linear term (-8). The two numbers that fit this criteria are -3 and -5, since the product of these two numbers is 15 and their sum is -8.

Step 4: Factor the quadratic expression

Now, we can rewrite the quadratic expression with its factors: $$ x^2-8x+15 = (x-3)(x-5) $$

Step 5: Combine the common factor with the factored quadratic expression

Finally, we can combine the factored expressions together: $$ x(x^2-8x+15) = x(x-3)(x-5) $$ So, the factored form of the trinomial $$x^{3}-8 x^{2}+15 x$$ is $$x(x-3)(x-5)$$.

Key Concepts

Common Factor

When factoring trinomials, the first step is to look for a common factor that is present in all the terms. In the trinomial given in the solution, which is \( x^3 - 8x^2 + 15x \), you’ll notice that each term has at least one \( x \). This is our common factor. By identifying and factoring out the common \( x \), you simplify the expression right at the start.

- **Why is finding a common factor important?** It simplifies the expression, making it easier to handle the rest of the problem. If you don't factor out early, you might miss a chance to simplify your work.

• **Easier Calculations**: Getting rid of the common factor means dealing with smaller numbers.
• **Reduces Complexity**: The expression looks less complicated once the common factor is out.

Recognizing and extracting the common factor is a critical skill in algebra and will aid in solving more complex problems later on.

Quadratic Expression

A quadratic expression is a polynomial of degree two. In the context of our factored trinomial \( x(x^2 - 8x + 15) \), the expression \( x^2 - 8x + 15 \) is a quadratic expression. This means it has the general form \( ax^2 + bx + c \).

There are some key points to keep in mind when dealing with quadratic expressions:

• **Standard Form**: Ensure your expression is in the form \( ax^2 + bx + c \) before proceeding with factoring.
• **Coefficient Roles**: The coefficient \( a \) affects whether the parabola opens up or down, while \( b \) affects where it is shifted along the x-axis, and \( c \) indicates the y-intercept.

In our example, \( x^2 - 8x + 15 \) is already neatly laid out for factoring, with \( a = 1, b = -8, \) and \( c = 15 \). This is crucial for finding the roots of the expression.

Factored Form

Factored form is an expression broken down into its simplest or most 'composable' form. For the given trinomial, this involves expressing it as a product of linear factors.

By breaking down the quadratic expression \( x^2 - 8x + 15 \) into two binomials, \((x-3)(x-5)\), we convert the expression into its factored form. This method involves determining the appropriate pairs of numbers that multiply to the constant term and add up to the coefficient of the linear term.

- **Why is it beneficial?**

• **Solving Equations**: It's easier to find solutions or roots when an expression is in factored form.
• **Graphing**: Constructing a graph is straightforward from the factored form as it provides the x-intercepts directly.

Factored form not only reduces complexity but also provides essential insight into the structure of the algebraic equation, helping find the zeroes quickly.

Sum and Product of Roots

The sum and product of roots method is a neat tool used in factoring quadratic equations. It helps find two numbers that both add up to a given sum and multiply to a specific product. In our exercise, we're focusing on the quadratic expression \( x^2 - 8x + 15 \).

- **Sum of Roots**: According to the problem, the roots must add up to \(-8\). This is derived from the linear coefficient \( b \) (from \(-bx\)).- **Product of Roots**: At the same time, the two numbers must multiply to \(15\), which matches the constant term \( c \).

• Number pair found in this context is \(-3\) and \(-5\).

This technique is particularly useful because it focuses directly on what is needed to split the middle term effectively, saving time and making calculations simpler. Through this method, we arrive at the binomial factors \((x-3)(x-5)\), leading to the final factored form.