Problem 43 - College Algebra

Question

In Exercises \(41-48,\) factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}-14 x+49$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x^{2}-14x+49\) is \((x-7)^{2}\)

Step 1: Identification

Identify that the given polynomial \(x^{2}-14x+49\) is in the form of \(a^{2}-2ab+b^{2}\), where \(a=x\), \(b=7\)

Step 2: Factorize

Factorize the polynomial by using the perfect square trinomial formula \(x^{2}-2ab+b^{2} = (a-b)^{2}\). Thus, \(x^{2}-14x+49 = (x-7)^{2}\)

Step 3: Write Down the Answer

Write down final factored form of the given polynomial which is \((x-7)^{2}\)

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. You'll often encounter perfect square trinomials in the form \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). These types of trinomials are particularly convenient to work with because they can be easily factored into a binomial squared.

For instance, consider the given polynomial: \(x^2 - 14x + 49\). It matches the pattern \(a^2 - 2ab + b^2 = (a-b)^2\), where \(a = x\) and \(b = 7\). By recognizing this pattern, we understand that this trinomial is a perfect square. Therefore, it can be factored as \((x-7)^2\).

Recognizing and factoring perfect square trinomials is a useful skill in algebra because it simplifies the polynomial into a much more manageable form.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operators such as addition, subtraction, multiplication, and division. Variables, such as \(x\) in our example, are symbols used to represent an unknown or varying number. They are a fundamental part of algebra, allowing for generalization and simplification of problems.

In the expression \(x^2 - 14x + 49\), \(x^2\) represents \(x\) multiplied by itself, \(-14x\) shows that \(x\) is multiplied by \(-14\), and \(49\) is a constant term. The beauty of algebraic expressions lies in how they enable us to solve complex problems by applying mathematical principles systematically.

• Variables represent unknowns or can stand for a set of values.
• Constants are fixed values.
• Coefficients indicate how many times to multiply the variable.

Understanding these components helps in manipulating the expressions for various mathematical operations such as solving equations or factoring.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler "factor" polynomials that, when multiplied together, yield the original polynomial. This process is akin to breaking down numbers into prime factors, making it easier to work with polynomials in equations.

In the problem of factoring \(x^2 - 14x + 49\), we used the recognition of a perfect square trinomial to factor it as \((x-7)^2\). Not all polynomials are this straightforward, and some might require different techniques such as the greatest common factor method, grouping, or applying the quadratic formula for factorization.

The essence of polynomial factorization lies in:

• Simplifying algebraic expressions.
• Finding solutions to equations.
• Understanding the behavior of polynomial graphs.

Effective factorization helps not only in solving equations but also in gaining deeper insights into the properties and characteristics of the polynomial function.