Problem 47 - Introductory Algebra for College Students

Question

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 trinomial \(x^{2}-14 x+49\) is \((x-7)^{2}\)

Step 1: Identify Potential Perfect Square Trinomial

A perfect square trinomial is in the form \((x-a)^{2}\) or equivalent to \(x^{2}-2ax+a^{2}\). The given expression is \(x^{2}-14x+49\). It can be written in the form \(x^{2}-2ax+a^{2}\) if the square root of 49 (which is 7) is equal to half of the absolute value of -14, (which is also 7). So, it's a potential perfect square trinomial.

Step 2: Factor the Perfect Square Trinomial

Since this trinomial meets the criteria of a perfect square trinomial, it can be factored into \((x-a)^{2}\), where a is half of the coefficient of the second term. Substituting the value we derived \(a=7\), we factor the trinomial to get \((x-7)^{2}\).

Step 3: Verify the Solution

Use the FOIL (First, Outer, Inner, Last) method to multiply out \((x-7)^{2}= (x-7)(x-7)\) to verify it. The result \(x^{2}-14x+49\) confirms the factoring is correct.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra that involves breaking down complex expressions into simpler factors. This process simplifies equations and makes them easier to work with. In our example, we deal with the trinomial \(x^{2} - 14x + 49\). Here, we first determine whether the expression is a perfect square trinomial.
To recognize a perfect square trinomial, we look for patterns that match the form \((x-a)^2 = x^2 - 2ax + a^2\). This involves:

• Identifying that the first term is a perfect square \((x^2)\).
• Verifying that the last term \(49\) is also a perfect square, where \(a^2 = 49\) implies \(a = 7\).
• Ensuring that the middle term \(-14x\) is twice the product of \(x\) and \(a\). In this case, \(-2 \times 7 \times x = -14x\), which confirms it.

Factoring then leads to \((x-7)^2\), representing the expression in its simplest terms. This skill of factoring is useful in solving quadratic equations, understanding polynomial functions, and performing algebraic manipulations.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, generally given by the form \(ax^2 + bx + c\). These expressions represent parabolas when graphed and have various properties that can be explored through different methods, such as factoring, completing the square, and using the quadratic formula.
In the case of a perfect square trinomial like \(x^2 - 14x + 49\), the quadratic can be factored neatly due to its specific pattern. This trinomial fits the perfect square form of \(x^2 - 2ax + a^2\), indicating it is the square of \((x-a)\). Recognizing perfect square trinomials is not only crucial for simplifying expressions but also for solving quadratic equations efficiently.
The ability to manipulate quadratic expressions helps in understanding:

• The vertex form of a parabola to find the vertex easily.
• The roots or solutions of the equation based on its factors.
• The relationship between different parts of the equation, such as \(a\), \(b\), and \(c\).

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables within them. They serve as shortcuts and tools for simplifying complex algebraic expressions. One of the most common identities is the square of a binomial, expressed as \((x-a)^2 = x^2 - 2ax + a^2\).
This identity is essential in recognizing and factoring perfect square trinomials. For instance, with \(x^2 - 14x + 49\):

• Identify that the trinomial follows \((x-a)^2\).
• Factor using the identity: resulting in \((x-7)^2\).

These identities also allow us to verify our work. After factoring, we can expand the expression using the FOIL method to check if it matches the original trinomial.
Utilizing algebraic identities not only speeds up calculations but also enhances understanding of how different algebraic components relate. They are foundational to more advanced topics in algebra, calculus, and beyond.