Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(90 v^{2}-181 v+90\)

Step-by-Step Solution

Answer

The factored form is \((9v - 10)(10v - 9)\).

Step 1: Identify a, b, and c

The quadratic polynomial is given as \(ax^2 + bx + c\). For the polynomial \(90v^2 - 181v + 90\), identify \(a = 90\), \(b = -181\), and \(c = 90\).

Step 2: Calculate the Discriminant

To determine the roots or factors of the quadratic equation, calculate the discriminant \(D = b^2 - 4ac\). Substitute \(b = -181\), \(a = 90\), and \(c = 90\):\[D = (-181)^2 - 4 \cdot 90 \cdot 90\]\\[D = 32761 - 32400 = 361\]

Step 3: Check for Factorability

Observe if the discriminant \(D = 361\) is a perfect square. Since \(361 = 19^2\), the polynomial can be factored using the quadratic formula or factoring technique.

Step 4: Use the Quadratic Formula

Since \(D = 361\) is a perfect square, use the quadratic formula to find roots:\[v = \frac{-b \pm \sqrt{D}}{2a}\]Substitute \(a = 90\), \(b = -181\), and \(D = 361\):\[v = \frac{181 \pm 19}{180}\]

Step 5: Solve for Roots

Calculate the two values for \(v\):1. \(v_1 = \frac{181 + 19}{180} = \frac{200}{180} = \frac{10}{9}\)2. \(v_2 = \frac{181 - 19}{180} = \frac{162}{180} = \frac{9}{10}\)

Step 6: Write Factored Form

Use the roots \(v_1\) and \(v_2\) to write the factored form of the polynomial:\[90v^2 - 181v + 90 = (9v - 10)(10v - 9)\]

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. The general form of a quadratic equation is \[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\). The presence of the \(x^2\) term differentiates it from linear equations.Quadratic equations can appear in various contexts such as physics, finance, and engineering, often representing curves like parabolas when graphed. Understanding how to solve these equations is crucial because it allows us to determine points of intersections, optimize values, and explore mathematical relationships. Quadratic equations can be solved by factoring, completing the square, using the quadratic formula, or graphing depending on the specific scenario and its complexity.

Discriminant

The discriminant is a specific value calculated from the coefficients of a quadratic equation. It is found using the formula:\[D = b^2 - 4ac\]The discriminant tells us important information about the nature of the roots of the quadratic equation:

• If \(D > 0\), there are two distinct real roots. If \(D\) is a perfect square, the solution can be found by factoring.
• If \(D = 0\), there is one real root, also known as a repeated or double root.
• If \(D < 0\), the equation has two complex roots, indicating no real intersection on the coordinate plane.

In the provided solution, the discriminant \(D\) is 361, a perfect square, leading to real roots that allow factoring.

Roots of Polynomial

Roots of a polynomial are the values of the variable that satisfy the equation, making it equal to zero. In simpler terms, they are where the graph of the polynomial touches or crosses the x-axis.For the quadratic equation \(90v^2 - 181v + 90 = 0\), the roots are calculated using the quadratic formula after identifying and computing the discriminant.In the exercise, these values are:\(v_1 = \frac{10}{9}\) and \(v_2 = \frac{9}{10}\). These roots mean if we plug these values back into the original polynomial, they will yield zero, confirming they are true solutions of the equation. Finding the roots allows factoring the polynomial further to the expression \((9v - 10)(10v - 9)\).

Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. The formula is expressed as:\[v = \frac{-b \pm \sqrt{D}}{2a}\]where \(D = b^2 - 4ac\) is the discriminant.This method is particularly useful when the quadratic is not easily factorable. It provides an exact solution for the roots, be they real or complex numbers.In the given problem, since the discriminant \(D = 361\) is a perfect square, the formula is used effectively to find roots:

• \( v_1 = \frac{181 + 19}{180} = \frac{10}{9}\)
• \( v_2 = \frac{181 - 19}{180} = \frac{9}{10}\)

Using the quadratic formula ensures a reliable solution for quadratic equations, especially when factoring is not straightforward.