Problem 41 - College Algebra

Question

In Exercises \(41-48,\) factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}+2 x+1$$

Step-by-Step Solution

Answer

The expression \( x^{2}+2x+1 \) is a perfect square trinomial and can be factored as \( (x+1)^2 \).

Step 1: Identifying numbers to be squared and multiplied

First, identify numbers that can be used to express the quadratic equation in the form \( (a+b)^{2} \). Here based on the form, 'a' could be \( x \) since \( x^2 \) exists in given equation which means 'a' is \( x \). Likewise 'b' should be a number when squared gives 1 and when multiplied by \( x \), gives 2x. The number 1 fits this criteria since \( 1^{2} = 1 \) and \( 2*x*1 = 2x \). Thus, 'b' is 1.

Step 2: Expressing the quadratic equation

Having figured out the values of 'a' and 'b', the equation can now be expressed in the form \( (a+b)^{2} \) or \( (x+1)^{2} \). Thus, the expression \(x^{2} + 2x + 1\) is a perfect square trinomial and can be factored as \( (x+1)^{2} \).

Key Concepts

Algebraic Factorization

Algebraic factorization is a process used to break down polynomials into simpler components or factors that, when multiplied together, give the original polynomial. This skill is fundamental in algebra and is particularly useful in simplifying equations and solving them more efficiently.

When factorizing algebraic expressions, especially quadratic expressions like \(x^{2}+bx+c\), the goal is to rewrite the expression as a product of two binomials. For example, if we have \(x^{2}+5x+6\), it can be factored into \(x+2\) times \(x+3\), since \(2*3=6\) and \(2+3=5\). This simplification can then assist in solving equations or finding zeroes of functions.

Practicing algebraic factorization helps students develop problem-solving skills, as it requires a mixture of critical thinking and recognition of patterns to deconstruct more complex expressions into their factors.

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that can be written as the square of a binomial. The general form is \(a^{2}+2ab+b^{2}\), where the trinomial equals \(a+b)^{2}\).

In essence, a perfect square trinomial is created when a binomial is multiplied by itself. For example, \(x+3)^{2}\) will expand to \(x^{2}+6x+9\), producing a perfect square trinomial. To identify these trinomials, look for the following signs:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

Understanding how to recognize and factor perfect square trinomials is vital for solving quadratic equations with ease and for simplifying algebraic expressions.

Quadratic Equations

Quadratic equations are second-degree polynomial equations that usually have the form \(ax^{2}+bx+c=0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. They are called 'quadratic' because 'quadra' refers to the square in Latin, and the highest power of the variable \(x\) is 2.

To solve these equations, one can use various methods such as factoring, completing the square, using the quadratic formula, or graphing. Factoring is often the quickest method when it's applicable, especially if the quadratic is a perfect square trinomial or can be easily decomposed into factors.

For instance, the perfect square trinomial \(x^{2}+2x+1\) can be factored to \(x+1)^{2}\), simplifying the solution process for \(x^{2}+2x+1=0\). Recognizing perfect square trinomials within quadratic equations and being able to factor them is a valuable skill in algebra.

Polynomial Factorization

Polynomial factorization involves dividing a polynomial into a product of simpler polynomials that, when multiplied together, give back the original polynomial. While quadratics are the most familiar polynomials to factor, the principles of factorization apply to polynomials of any degree.

Factoring methods may include pulling out a greatest common factor, grouping, using special products such as perfect square trinomials and difference of squares, or applying the factor theorem and synthetic division for higher-degree polynomials.

Understanding the different methods suitable for polynomial factorization is crucial, as it allows for the simplification of complex algebraic expressions and enables one to solve polynomial equations more easily. Mastering polynomial factorization can offer students the tools to tackle a wide variety of algebraic challenges.