Problem 19 - College Algebra

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

Step 1: Move All Terms to One Side

Subtract the terms on the right side of the equation from both sides to set the equation to zero:\[ 5x^3 + 45x - 2x^2 - 18 = 0 \]

Step 2: Rearrange the Terms

Rearrange the terms in descending order of powers of \( x \): \[ 5x^3 - 2x^2 + 45x - 18 = 0 \]

Step 3: Group Terms and Factor Common Factors

Group terms to factor by grouping. Group the first two and the last two terms:\[ (5x^3 - 2x^2) + (45x - 18) = 0 \]Factor out common factors from each group:\[ x^2(5x - 2) + 9(5x - 2) = 0 \]

Step 4: Factor Out the Common Binomial

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

Step 5: Solve Each Factor

Set each factor to zero and solve for \( x \):For \( x^2 + 9 = 0 \):\[ x^2 = -9 \]This gives no real solutions since negative numbers do not have real square roots.For \( 5x - 2 = 0 \):\[ 5x = 2 \]\[ x = \frac{2}{5} \]

Step 6: Conclusion

The only real solution to the polynomial equation is \( x = \frac{2}{5} \).

Key Concepts

Factoring

Factoring is one of the primary techniques used in solving polynomial equations. It involves expressing a polynomial as a product of its factors, which are simpler polynomials, often linear or quadratic in nature. This approach breaks down complex equations into more manageable parts, allowing us to leverage properties like the zero-product property. The zero-product property states that if the product of several factors is zero, at least one of the factors must be zero.

In our given exercise, the equation was re-arranged so we could apply factoring effectively. First, terms on one side of the equation were collected to equal zero, then arranged in decreasing powers of the variable, which is a common strategy to simplify the equation.

Factoring involves careful observation to identify common factors among terms, as seen in the grouping technique used here. Recognizing patterns, such as common binomials that can be factored out, further simplifies solving polynomials. Developing a keen eye for these factors allows you to solve polynomial equations efficiently and correctly.

Grouping

Grouping is a strategic approach for factoring that involves organizing terms with similar characteristics together. This technique is particularly useful for polynomials where straightforward factoring is not immediately possible. By grouping terms, you prepare the polynomial for factorization by identifying and isolating common factors.

In practice, as seen in the original exercise, terms of the polynomial are grouped into pairs. For instance, in the equation \[ 5x^3 - 2x^2 + 45x - 18 = 0 \] terms were grouped as follows:

• First Group: \(5x^3 - 2x^2\)
• Second Group: \(45x - 18\)

By doing this, common factors within each group can be identified. Each group is then factored separately, which can sometimes reveal a common binomial factor across the entire expression, allowing for further simplification.

In our example, this method made it easier to spot the common binomial, \((5x - 2)\), allowing us to reduce the polynomial into a more solvable form. Grouping is a valuable skill, turning a complicated polynomial into a sequence of simpler tasks.

Real Solutions

Real solutions to polynomial equations are the solutions that are observable within the set of real numbers. These are the solutions without imaginary components and can be plotted on the real number line.

In the original equation \[ (x^2 + 9)(5x - 2) = 0 \] our task was to find the real solutions. By setting each factor equal to zero, \[ x^2 + 9 = 0 \] and \[ 5x - 2 = 0 \] we proceed with solving for \(x\):

• \(x^2 + 9 = 0\) resulted in no real solutions because square roots of negative numbers are not real.
• \(5x - 2 = 0\) gives a real solution of \(x = \frac{2}{5}\).

The only real solution provides us with tangible interpretations in many practical applications. Understanding which solutions are real enables us to solve real-world problems effectively, focusing on applicable solutions that fit within the realm of reality.