Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$(x+3)^{3}-(3 x-1)^{2}=x^{3}+4$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{2}{3}\).

Step 1: Expand the Left Hand Side

The given equation is \((x+3)^3 - (3x-1)^2 = x^3 + 4\). First, we'll expand the terms on the left-hand side. For \((x + 3)^3\), we get:\[ (x+3)^3 = x^3 + 9x^2 + 27x + 27 \]For \((3x-1)^2\), we have:\[ (3x-1)^2 = 9x^2 - 6x + 1 \]So the left hand side becomes:\[ (x^3 + 9x^2 + 27x + 27) - (9x^2 - 6x + 1) = x^3 + 9x^2 + 27x + 27 - 9x^2 + 6x - 1 \]

Step 2: Simplify the Left Hand Side

Simplify the expression we obtained:\[ x^3 + 9x^2 + 27x + 27 - 9x^2 + 6x - 1 = x^3 + 33x + 26 \]

Step 3: Setup the Simplified Equation

Insert the simplified expression back into the equation. We now have:\[ x^3 + 33x + 26 = x^3 + 4 \]

Step 4: Isolate the Variable

Subtract \(x^3\) from both sides of the equation to eliminate the \(x^3\) terms:\[ 33x + 26 = 4 \]

Step 5: Solve for x

Subtract 26 from both sides of the equation:\[ 33x + 26 - 26 = 4 - 26 \]\[ 33x = -22 \]Divide both sides by 33 to solve for \(x\):\[ x = \frac{-22}{33} = -\frac{2}{3} \]

Key Concepts

Polynomial Expansion

Polynomial expansion is a process where we take an expression raised to a power and rewrite it as a sum of terms without exponents. This is often done using binomial expansions, like in our example where we expanded

• \((x + 3)^3\) into \(x^3 + 9x^2 + 27x + 27\)
• \((3x - 1)^2\) into \(9x^2 - 6x + 1\)

The expansions make complex equations simpler to work with by showing every constituent part. This is crucial before undertaking other operations such as combining like terms or simplifying the equation. As you become comfortable with expansions, recognizing patterns and using formulas can speed up the process.

Algebraic Simplification

Algebraic simplification involves combining and reducing expressions to make them easier to work with. After expanding polynomials, like those in the current problem, you'll need to simplify by combining like terms. For example:

• Combining terms like \(9x^2\) and \(-9x^2\) simplifies the expression dramatically.
• The original expanded expression \(x^3 + 9x^2 + 27x + 27 - 9x^2 + 6x - 1\) simplifies to \(x^3 + 33x + 26\).

When simplifying, it helps to carefully track each term. Look out for common variables and constants in your expression that can be combined. This procedure not only reduces complexity but also prepares the equation for further analysis or solving for variables.

Variable Isolation

Variable isolation is the method of rearranging an equation so that one variable stands alone on one side of the equation. This makes it possible to solve for that variable directly. In our example, after simplifying:

• We subtracted \(x^3\) from both sides, leaving \(33x + 26 = 4\).
• Then subtracted 26 from both sides, resulting in \(33x = -22\).

The purpose of these steps is to get a single occurrence of \(x\) on one side of the equation, allowing for straightforward solutions. All inverse operations (like subtraction when we see addition) aim at pushing terms away from the variable we're solving for.

Polynomial Equation

A polynomial equation is a mathematical expression containing variables and coefficients, involving terms in the form of \(ax^n\) where \(n\) is a non-negative integer. In solving polynomial equations, like the original problem \((x+3)^3-(3x-1)^2=x^3+4\), the goal is to find the values of the variable that make the equation true.

• Such equations can range from simple linear forms to complex cubic expressions, each demanding a unique approach.
• For equations that are polynomials, recognizing the degree helps set the approach. That includes expansions, simplifications, and isolations as necessary.

Solving these equations might require various algebraic methods, including factoring or using the quadratic formula, especially for polynomials of higher degree.