Problem 45 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ x^{2}-8 x+15 $$

Step-by-Step Solution

Answer

The trinomial is factored as \((x - 3)(x - 5)\).

Step 1: Identify the trinomial structure

The given trinomial is in the form of \(ax^2 + bx + c\). Here, \(a = 1\), \(b = -8\), and \(c = 15\). Since there is no greatest common factor (GCF) other than 1 between the terms, we proceed to factoring the trinomial.

Step 2: Find factors of the constant term

We need to find two numbers that multiply to give \(c = 15\) and add to give \(b = -8\). The factor pairs of 15 are (1, 15), (3, 5), (-1, -15), and (-3, -5).

Step 3: Select the correct factor pair

We need these numbers to also add up to \(b = -8\). Out of the factor pairs, \((-3) + (-5) = -8\) is the pair that satisfies this condition.

Step 4: Write the factors of the trinomial

Using the factor pair \(-3\) and \(-5\), we can write the expression as \((x - 3)(x - 5)\).

Step 5: Verify the solution

Expand \((x - 3)(x - 5)\) to ensure it equals the original trinomial. \((x - 3)(x - 5) = x^2 - 5x - 3x + 15 = x^2 - 8x + 15\), which matches the given trinomial.

Key Concepts

Polynomials

Polynomials are mathematical expressions made up of variables and constants combined using only addition, subtraction, and multiplication. These expressions can have terms with a non-negative integer exponent. A typical polynomial looks like this:

• Example: \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

Here, each term is composed of a coefficient \(a_n\) and a variable \(x\) raised to an exponent. The degree of a polynomial is determined by the highest exponent it contains.
Polynomials can come in various forms such as monomials, binomials, and trinomials, depending on the number of terms they have. They play a fundamental role in algebra and are utilized in various mathematical operations, including factoring.

Quadratic Expressions

A quadratic expression is a specific type of polynomial that includes terms up to the second degree. It typically follows the form:

• \(ax^2 + bx + c\)

Where:

• \(a, b,\) and \(c\) are constants.
• \(x\) is the variable, and the highest power of \(x\) is 2.

Quadratic expressions can often be factored to simplify solutions or find roots. In the context of this exercise, you dealt with a quadratic expression \(x^2 - 8x + 15\). Factoring these expressions involves breaking them down into the product of two binomials or identifying their roots.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor shared by all terms of a polynomial. When factoring polynomials, finding the GCF is an essential first step.
The GCF is particularly helpful because:

• It simplifies the polynomial, making other factoring steps easier.
• It ensures that any common factors are accounted for in the final factored form.

In some exercises, the first step is to determine if there is a GCF. If a GCF is present, it is factored out before proceeding with further strategies, such as splitting the middle term of a quadratic expression or other methods.
In our original exercise, the trinomial had no GCF other than 1, so we directly proceeded to factor the expression without this step.

Factorization Steps

Factorization is the process of rewriting a polynomial as a product of its factors. For quadratic expressions like the one in the exercise \(x^2 - 8x + 15\), the main goal is to express it as a product of binomials.
Here are the general steps involved:

• Identify the quadratic structure: Recognize the form \(ax^2 + bx + c\).
• Check for a GCF: This simplifies later steps.
• Determine factor pairs of \(c\): For the trinomial \(x^2 - 8x + 15\), find pairs that multiply to \(15\).
• Select the correct pair: Choose the pair that also adds up to \(b = -8\). Here, (-3) and (-5) fit the bill.
• Re-write the expression: Use the chosen pair to factor the trinomial as \((x-3)(x-5)\).
• Verify: Expand the factors back to ensure they produce the original quadratic.

Factorization not only simplifies expressions but also aids in solving equations and understanding the roots of quadratic expressions.