Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, solve the following polynomial equations by grouping and factoring. $$ 5 x^{3}+45 x=2 x^{2}+18 $$

Step-by-Step Solution

Answer

The real solution is \(x = \frac{2}{5}\).

Step 1: Set the equation to zero

Start by moving all terms to one side of the equation to set it equal to zero. The given equation is \(5x^3 + 45x = 2x^2 + 18\). Subtract \(2x^2 + 18\) from both sides to obtain: \[5x^3 + 45x - 2x^2 - 18 = 0.\]

Step 2: Rearrange and group terms

Rearrange the terms in the equation to facilitate grouping: \[5x^3 - 2x^2 + 45x - 18 = 0.\]Now, group the first two terms together and the last two terms together:\[(5x^3 - 2x^2) + (45x - 18) = 0.\]

Step 3: Factor by grouping

Factor out the greatest common factor from each group separately. For the first group \((5x^3 - 2x^2)\), factor out \(x^2\): \[x^2(5x - 2).\]For the second group \((45x - 18)\), factor out \(9\):\[9(5x - 2).\]So the equation becomes: \[x^2(5x - 2) + 9(5x - 2) = 0.\]

Step 4: Factor out the common factor

Notice that \((5x - 2)\) is a common factor. Factor it out:\[(5x - 2)(x^2 + 9) = 0.\]

Step 5: Solve for the roots

Set each factor equal to zero and solve for \(x\):1. \(5x - 2 = 0\): Solve for \(x\) to get \(x = \frac{2}{5}\). 2. \(x^2 + 9 = 0\): Solve for \(x\) to get \(x^2 = -9\), which implies \(x = 3i\) or \(x = -3i\). Since negative roots lead to imaginary numbers, ignore them if you're looking for real solutions.

Key Concepts

Grouping and Factoring

Grouping and factoring is a crucial skill when solving polynomial equations. It involves rearranging and organizing terms to make the equation easier to solve. For example, in a polynomial like \(5x^3 - 2x^2 + 45x - 18 = 0\), the idea is to identify terms that can be grouped together. Here, grouping \(5x^3 - 2x^2\) and \(45x - 18\) separately allows us to factor each group more efficiently.

• Identify common factors within each group.
• Factor out these common factors.
• Look for common terms between the groups after factoring.

In the example, factoring \(x^2\) out of \(5x^3 - 2x^2\) gives \(x^2(5x - 2)\), and factoring 9 out of \(45x - 18\) gives \(9(5x - 2)\). Notice that both groups share the common factor \((5x - 2)\), which can then be factored out, resulting in \((5x - 2)(x^2 + 9) = 0\). This method streamlines the process of solving complex polynomial equations.

Solving Equations

Solving polynomial equations involves finding the values of \(x\) that satisfy the equation. Once the equation has been factored, as in our example \((5x - 2)(x^2 + 9) = 0\), the next step is to set each factor equal to zero.

• Each factor \(a\) and \(b\) in the equation \(a \cdot b = 0\) must individually equal zero.
• Setting \(5x - 2 = 0\) leads to the solution \(x = \frac{2}{5}\).
• Setting \(x^2 + 9 = 0\) leads to solutions involving complex numbers, as \(x^2 = -9\).

This process, called "zero-product property," allows us to determine all potential solutions of the polynomial. It simplifies complex problems by breaking them into simpler, manageable parts that can be solved separately.

Complex Numbers

Complex numbers come into play when solving polynomial equations that lead to negative square roots, such as \(x^2 = -9\). This introduces the imaginary unit \(i\), where \(i^2 = -1\).Using this concept, we can express square roots of negative numbers. So, when solving \(x^2 + 9 = 0\), we rearrange to find \(x^2 = -9\). This gives \(x = \sqrt{-9}\), which is simplified using the imaginary unit:

• The expression \(x = \pm 3i\) describes two possible roots: \(x = 3i\) and \(x = -3i\).
• These are called "complex roots" due to their involvement of the imaginary unit.

Understanding complex numbers is vital for fully solving certain polynomial equations, especially those that do not have real solutions.

Roots of Polynomials

The roots of a polynomial are the values of \(x\) that make the equation true, i.e., solutions to the polynomial equation. In our example equation \((5x - 2)(x^2 + 9) = 0\), we derived three roots: one real and two complex:

• \(x = \frac{2}{5}\): a real root.
• \(x = 3i\): a complex root.
• \(x = -3i\): another complex root.

Knowing your polynomial's degree is helpful. A degree-three polynomial, like this example, suggests up to three roots are possible. Each root provides insight into the behavior of the polynomial. Complex roots often occur in conjugate pairs, indicating symmetrical behavior along the imaginary axis.Understanding the concept of roots, and distinguishing between real and complex roots, is critical in polynomial algebra's realm.