Problem 64 - College Algebra

Question

Give an example of a polynomial equation that has no real roots. Describe how you obtained the equation.

Step-by-Step Solution

Answer

An example of a polynomial equation that has no real roots is \(x^2 + 2x + 3 = 0\). This was obtained by selecting 1, 2 and 3 as coefficients for a, b and c respectively, to form a quadratic equation. By calculating discrimant we ensured that it is negative which means it has no real roots.

Step 1: Choosing a coefficients for the quadratic equation

Consider the general quadratic equation \(ax^2 + bx + c = 0\). Choose any values for a, b, c. For example, \(a = 1\), \(b = 2\), \(c = 3\) to form the equation \(x^2 + 2x + 3 = 0\).

Step 2: Confirming the absence of real roots

Substitute the chosen coefficients into the discriminant formula \(D = b^2 - 4ac\). For \(a = 1\), \(b = 2\), and \(c = 3\), we get \(D = 2^2 - 4*1*3 = 4 - 12 = -8\). The negative value of the discriminant confirms that the quadratic equation has no real roots.

Key Concepts

Complex Roots

Consider polynomial equations as puzzles where roots are the pieces that fit perfectly into the setup of the equation, giving us the 'solution'. With real numbers, our number line is limited, but math takes us beyond that—with complex roots, we enter a realm where even the seemingly unsolvable gets a solution.

In the exercise provided, we know that the equation x^2 + 2x + 3 = 0 doesn’t have any solutions on the usual number line. But that doesn't mean there are no solutions. They just happen to be complex. By encountering a negative discriminant, we're nudged towards these complex solutions. They are typically written in the form a + bi, where a and b are real numbers, and i represents the square root of -1, making it possible for us to solve equations that have no real roots.

Discriminant

The discriminant is a powerful tool in algebra. It determines the nature of the roots of a quadratic equation without actually solving the equation. It's represented by the symbol D and is part of the quadratic formula.

The formula for the discriminant is D = b^2 - 4ac. Here's a quick guide to interpreting its value:

• If D > 0, there are two distinct real roots.
• If D = 0, there is exactly one real root (also known as a repeated or double root).
• If D < 0, there are no real roots, but instead, there are two complex roots.

Returning to our example, by plugging in the coefficients a = 1, b = 2, and c = 3 into the formula, we calculated the discriminant as -8. Since this is less than zero, it confirms that we won't find our answers on the real number line, hinting at the presence of complex solutions.

Quadratic Equations

Laying at the heart of our example is a quadratic equation, recognizable by its highest exponent being a square (x^2). Quadratic equations look like this: ax^2 + bx + c = 0. The letters a, b, and c are placeholders for any real number, with a ≠ 0. These equations often model real-world phenomena like projectile motion or market economics.

A quadratic equation can be solved by various methods: factoring, completing the square, using the quadratic formula, or graphing. Each method has its merits and picking the right one depends on the specific nature of the equation we're faced with. The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), is particularly helpful as it provides a direct way to find the roots of any quadratic equation, provided you compute the discriminant correctly, revealing whether the roots are real or complex.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. They are the building blocks of equations. In algebra, we manipulate these expressions to solve equations and find unknown values. An example of an algebraic expression is 2x + 3, which represents a number that is three more than twice another number.

Expressions become increasingly complicated with the addition of exponents and coefficients, as in our quadratic example, x^2 + 2x + 3. This particular expression doesn't represent merely a line, but a curve when graphed. Manipulating algebraic expressions involves understanding the rules of operation and being able to apply them to simplify or expand these expressions, which is a fundamental skill in solving more advanced problems in mathematics.