Problem 12 Factor each trinomial completely... [FREE SOLUTION]

Chapter 6: Problem 12

Factor each trinomial completely. If a polynomial can't be factored, write "prime." See Examples I through 8 . $$ x^{2}-7 x+5 $$

Short Answer

Expert verified

The polynomial is prime.

Step by step solution

Identify coefficients

The trinomial is in the standard form of a quadratic equation: $ ax^2 + bx + c $, where $ a = 1 $, $ b = -7 $, and $ c = 5 $.

Find two numbers that multiply to a*c and add to b

We want to find two numbers that multiply to $ a \cdot c = 1 \cdot 5 = 5 $ and add to $ b = -7 $. The possible pairs are $(1, 5)$, $(-1, -5)$, $(5, 1)$, $(-5, -1)$, $(7, -7)$, etc. However, no pair sums to $-7$.

Check if trinomial is factorable

Since no two integers multiply to 5 and add to -7, the trinomial $ x^2 - 7x + 5 $ cannot be factored over the integers.

Conclude trinomial is prime

As we could not find integers that satisfy the needed conditions, the polynomial $ x^2 - 7x + 5 $ is prime.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Trinomials

Quadratic trinomials are polynomials that consist of three terms and involve a variable squared, a variable raised to the first power, and a constant term. In mathematical language, they are written as $ ax^2 + bx + c $. Here, $ a $, $ b $, and $ c $ are coefficients and $ x $ is the variable.These trinomials represent a fundamental type of polynomial expressions, commonly encountered in algebra.

Quadratic: The highest exponent of the variable is 2.
Trinomial: Contains three terms.

Let's take the example of $ x^2 - 7x + 5 $. This is a quadratic trinomial because it follows the format $ ax^2 + bx + c $. The challenge is often to factor these expressions into products of simpler binomials, whenever possible.

Standard Form Quadratic

The standard form of a quadratic equation is a specific way of writing quadratic expressions. It takes the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants with $ a eq 0 $.Here's why the standard form is useful:

It easily allows for application of the quadratic formula or completing the square.
It provides a straightforward method for graphing parabolas.
It simplifies the process of determining roots, intercepts, and the vertex of the parabola.

In our example $ x^2 - 7x + 5 $, it is already in standard form with $ a=1 $, $ b=-7 $, and $ c=5 $. Recognizing this allows us to use methods like factoring, the quadratic formula, or completing the square to solve it or determine its characteristics.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored into the product of polynomials of degree lower than itself with integer coefficients.Much like how prime numbers are numbers only divisible by 1 and themselves, prime polynomials cannot be broken down into simpler polynomials using integer coefficients alone.Reasons why a polynomial might be prime include:

The coefficients do not allow for factorization over integers.
All attempts to find suitable factors result in non-integer solutions.

With our trinomial $ x^2 - 7x + 5 $, we can't find any integer factor pairs for the constant term $ 5 $ that add up to $ -7 $. Thus, it is considered a prime polynomial. It's an important concept as it tells us when a polynomial can't be simplified further by standard factoring methods.

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91影视

Factor each trinomial completely. If a polynomial can't be factored, write "prime." See Examples I through 8 . $$ x^{2}-7 x+5 $$

Short Answer

Step by step solution

Identify coefficients

Find two numbers that multiply to a*c and add to b

Check if trinomial is factorable

Conclude trinomial is prime

Key Concepts

Quadratic Trinomials

Standard Form Quadratic

Prime Polynomial

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