Problem 12 - Algebra A Combined Function

Question

Factor each trinomial completely. If a polynomial can't be factored, write "prime." $$ x^{2}-7 x+5 $$

Step-by-Step Solution

Answer

The polynomial is prime and cannot be factored using integers.

Step 1: Analyze the Trinomial

Identify the trinomial in the form of \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = -7 \), and \( c = 5 \). We need to find two numbers that multiply to \( a \times c = 1 \times 5 = 5 \) and add to \( -7 \).

Step 2: Check Factor Pairs

List factor pairs of \( 5 \): \( (1, 5) \) and \((-1, -5)\). None of these pairs add up to \(-7\). So, this trinomial cannot be factored using integers.

Step 3: Conclude Polynomial Status

Since no integer pairs satisfy both conditions for factoring, \( x^2 - 7x + 5 \) is a prime polynomial.

Key Concepts

Understanding Trinomials

A trinomial is a type of polynomial that consists of exactly three terms. Consider it like a math sentence with three pieces connected through plus or minus signs. In many cases, trinomials commonly appear in the form \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) is the variable.

For example, in the trinomial \( x^2 - 7x + 5 \):

• \( a = 1 \)
• \( b = -7 \)
• \( c = 5 \)

Each of these coefficients plays a role in how the polynomial can be factored (if at all). The aim is often to express the trinomial as a product of two binomials, simplifying calculations and solutions in other equations. Learning to work with trinomials is important because they appear often, particularly in quadratic equations.

Prime Polynomial Characteristics

A prime polynomial, like a prime number, cannot be broken down into simpler components through factoring. It means there are no pairs of integers that satisfy the factorization conditions to transform it into a multiplication of simpler polynomials.

For example, when we examined the polynomial \( x^2 - 7x + 5 \), we realized it cannot be factored since no integer pairs of its product \( a \times c = 5 \) add up to \( b = -7 \).

Therefore, this polynomial stays in its simplest form. Recognizing when a polynomial is prime is crucial because it informs how you approach solving related equations and suggests that other techniques may be needed to find roots or solutions.

Integer Factorization

Integer factorization involves breaking down numbers into their simplest "building blocks," which are prime numbers. When it comes to polynomials, integer factorization means looking for pairs of integers that satisfy specific conditions—like a given product and sum.

In the context of a quadratic trinomial \( ax^2 + bx + c \), you aim to find two numbers that:

• Multiply to \( a \times c \)
• Add to \( b \)

This factoring strategy simplifies the trinomial into two binomials.

However, there are cases, such as with \( x^2 - 7x + 5 \), where no such integers exist, leading us to label the polynomial as prime. Understanding integer factorization aids in grasping broader mathematical concepts and problem-solving strategies, as it often serves as a necessary step to simplify complex equations.