Understanding and Solving the Product of (2x - 32x^5)

The expression (2x - 32x^5) is a combination of algebraic terms and can be broken down into simpler components using the FOIL method (First, Outside, Inside, Last) or the distributive property. This article will guide you through the process of solving the product of (2x - 32x^5).

Introduction to the FOIL Method and Distributive Property

The FOIL method is a technique used to multiply two binomials. However, in this case, we are working with a single expression. The distributive property, on the other hand, can be extended to any number of terms. For the expression (2x - 32x^5), we can treat it as (2x) - (32x^5) and distribute it appropriately.

Step-by-Step Solution

Using the Distributive Property

To find the product, we can use the distributive property to expand the terms:

2x - 32x^5 2x times; 2x - 2x times; 5 - 32x^5 times; 2x 32x^5 times; 5

However, since there is no second binomial to multiply by, we simplify it as follows:

Combining the terms, we get:

4x^2 - 1 - 64x^6 - 16^5

Using the FOIL Method for Binomials

Though the expression is not a binomial, we can still demonstrate the technique for similar problems. For example, if the expression was (2x 5 - 32x^5 3x), we could apply the FOIL method more comprehensively:

(2x 5 - 32x^5 3x) 2x times; 2x 2x times; 5 2x times; 3x 2x times; -32x^5 5 times; 2x 5 times; 5 5 times; 3x 5 times; -32x^5 -32x^5 times; 2x -32x^5 times; 5 -32x^5 times; 3x -32x^5 times; -32x^5 3x times; 2x 3x times; 5 3x times; 3x 3x times; -32x^5

Simplifying each term and combining like terms, we get a more complex result. Here, we focus on the given expression and its simplified form:

Final Simplified Expression

The simplified product of (2x - 32x^5) is:

4x^2 - 4x - 15

Solving for Specific Values of x

To solve for specific values of x, we can set the simplified expression equal to zero and solve for x:

4x^2 - 4x - 15 0

Using the quadratic formula (x frac{-b pm sqrt{b^2 - 4ac}}{2a}) where (a 4), (b -4), and (c -15), we get:

x (frac{-(-4) pm sqrt{(-4)^2 - 4 cdot 4 cdot (-15)}}{2 cdot 4})

x (frac{4 pm sqrt{16 240}}{8})

x (frac{4 pm sqrt{256}}{8})

x (frac{4 pm 16}{8})

x (frac{20}{8}) or x (frac{-12}{8})

x 2.5 or x -1.5

Thus, the solutions are:

x 1.5 or x -2.5

Conclusion

The product of (2x - 32x^5) is 4x^2 - 4x - 15. This solution can be derived using the distributive property or the FOIL method, depending on the complexity of the expression. Understanding these techniques is essential for solving more complex algebraic expressions.