Problem 16 - Algebra Form and Function

Question

Give all the solutions of the equations. $$ (u+3)^{3}=-(u+3)^{3} $$

Step-by-Step Solution

Answer

Answer: The solution is \(u = -3\).

Step 1: Set up the equation

Given equation: \((u+3)^3 = -(u+3)^3\)

Step 2: Add \((u+3)^3\) to both sides of the equation

To remove the negative sign on the right side of the equation, we will add \((u+3)^3\) to both sides: \((u+3)^3 + (u+3)^3 = 0\)

Step 3: Factor out \((u+3)^3\) from both sides

We can factor out \((u+3)^3\) from each side: \((u+3)^3(1+1) = 0\)

Step 4: Simplify the expression

We have: \((u+3)^3(2) = 0\)

Step 5: Divide both sides by 2 and solve for u

To find the value of u that results in the expression being equal to zero, we can divide both sides of the equation by 2: \((u+3)^3 = 0\) Now, to find the value of u, we can take the cube root of both sides: \(u+3 = 0\) And finally, we can subtract 3 from both sides to solve for u: \(u = -3\)

Step 6: State the solution

The only solution to the given equation is \(u = -3\).

Key Concepts

Factoring

Factoring is a powerful tool in solving equations, especially when dealing with expressions raised to a power.
In the step-by-step solution provided, factoring plays a crucial role in simplifying and solving the equation. The original equation was \((u+3)^3 + (u+3)^3 = 0\).
By factoring out \((u+3)^3\) from each term, we simplify the expression significantly to:

• \((u+3)^3(2) = 0\)

This step makes solving for variables much easier because we're left with a linear expression. Once you are able to express the entire equation as a product equal to zero, you can use the Zero Product Property. This property states that if two numbers multiply to give zero, at least one of them must be zero. In mathematical terms, if \(ab = 0,\) then \(a = 0\) or \(b = 0\).
In our case, since \((u+3)^3\) is the product involving the variable \(u\), setting \((u+3)^3 = 0\) is what helps us solve for \(u\). Figuring out when any factor equals zero simplifies our equation to its most fundamental component. This makes it much easier to solve.

Polynomials

Polynomials are mathematical expressions consisting of variables raised to different powers and their coefficients.
In the given exercise, we see a cubic polynomial, \((u+3)^3\), indicating that it is raised to the third power. Such polynomials can have up to three solutions or roots, depending upon how they are factored and equated to zero.
This specific example is straightforward because we are dealing with the equation equated to a negative version of itself, making some terms cancel out during simplification.
When working with polynomials, you often want to simplify them to a basic form, like factored or standard form. Here, knowing that you can simplify the original expression to

• \((u+3)^3(2) = 0\),

helps in finding the solution because it reduces a complex polynomial expression to something much easier to work with. In many cases, polynomials are solved by approximation methods, factorization, or using the quadratic formula when applicable, but here, factoring was also effective to directly find solutions.

Cube Root

The concept of a cube root is essential in this exercise's solution; it helps isolate the value of \(u\).
Once we reduced the expression to \((u+3)^3 = 0\), taking the cube root is necessary to simplify further. The cube root is the inverse operation of cubing a number.
It means finding a number which, when multiplied by itself three times, results in the original number. When we have \((u+3)^3 = 0\), we take the cube root of both sides:

• \(u+3 = \sqrt[3]{0}\)
• o \(u+3 = 0\)

The result is a simple linear equation easily solved by isolating \(u\) through simple arithmetic: subtracting 3 from both sides gives \(u = -3\). Keep in mind, taking the cube root is generally valid for all real numbers, including zero and negative numbers.Understanding cube roots is crucial in algebra, especially when solving polynomials, as it allows you to revert a cubed expression back to its basic form.