Problem 22 - College Algebra

Question

Factor each trinomial, or state that the trinomial is prime. $$ x^{2}-14 x+45 $$

Step-by-Step Solution

Answer

The factored form of the given trinomial \(x^{2}-14x+45\) is \((x - 9)(x - 5)\).

Step 1: Identifying the Trinomial

The given trinomial is \(x^{2}-14x+45\). The general form of a trinomial is \(ax^{2} + bx + c\), where \(a\), \(b\) and \(c\) are constants.

Step 2: Checking if the Trinomial can be Factored

If there exist two numbers \(m\) and \(n\) such that they add up to '-14' (middle term coefficient of the quadratic equation) and when multiplied together give '45' (the last term of the quadratic equation), then the equation can be factored.

Step 3: Finding the numbers

The two numbers that meet these conditions are -9 and -5. That is because, -9 + -5 equals -14 and -9 * -5 equals 45.

Step 4: Factorizing the Trinomial

The two factors of the quadratic equation, using the numbers found in the previous step, are given by \((x - 9)(x - 5)\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents the variable. Understanding quadratic equations is fundamental in algebra as it allows us to find the values of \(x\) that satisfy the equation, commonly known as roots or solutions of the equation.

To solve a quadratic equation, you can use several methods including:

• Factoring, which involves expressing the quadratic equation as a product of binomials if possible.
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which can always give a solution if you cannot factor the trinomial.
• Completing the square, which involves rewriting the equation in a perfect square form to find the roots.

In the exercise, factoring was used to solve the quadratic equation by identifying two numbers, \(-9\) and \(-5\), that both add and multiply to the coefficients \(b\) and \(c\) respectively.

Trinomial

In algebra, a trinomial is an expression that contains three terms. Typically, trinomial expressions have the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. Understanding their structure is essential for manipulating and factoring them appropriately.

Factoring trinomials involves writing the expression as the product of two binomials. This process simplifies the expression and is useful in solving quadratic equations. The key to factoring trinomials is finding two numbers that:

• Multiply together to give the third term \(c\).
• Add up to give the middle coefficient \(b\).

For example, in the trinomial \(x^2 - 14x + 45\), our goal is to break it into \((x + m)(x + n)\) such that \(m\) and \(n\) sum to \(-14\) and multiply to \(45\). In this case, the correct pair of numbers is \(-9\) and \(-5\).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They form the foundation of algebra and are used to represent real-world quantities in a simplified manner. In algebra, you work with things like monomials, binomials, and trinomials, each named for the number of terms they contain.

Trinomials, like those found in quadratic equations, are an example of algebraic expressions. When factoring trinomials, you are essentially manipulating the expression into simpler parts or binomials, which makes the equation easier to solve or analyze.

• Algebraic expressions can represent complex relationships and allow for simplification when factoring.
• They enable easier computation and problem-solving with known algebraic techniques.

The ability to factor and simplify expressions is crucial in algebra, aiding in solving equations and making sense of complex mathematical problems. As demonstrated in the exercise, factoring helps break down the expression \(x^2 - 14x + 45\) into simpler binomials such as \((x - 9)(x - 5)\).