Problem 8 - Algebra Form and Function

Question

Give all the solutions of the equations. $$ (x+3)\left(1-x^{2}\right)=0 $$

Step-by-Step Solution

Answer

Question: Find all the solutions for the given equation $$(x+3)\left(1-x^{2}\right)=0$$. Answer: The solutions for the equation are $$x = -3, 1, -1$$.

Step 1: Identify the factors

The given equation is a product of two factors, $$(x + 3)$$ and $$(1 - x^{2})$$, which are equal to zero.

Step 2: Set each factor to zero

We'll solve for $$x$$ in each factor when they are set to zero: 1. $$(x + 3) = 0$$ 2. $$(1 - x^{2}) = 0$$

Step 3: Solve for x in each factor

For factor 1: $$x + 3 = 0$$, subtract 3 from both sides to get $$x = -3$$ For factor 2: $$1 - x^{2} = 0$$, add $$x^{2}$$ to both sides to get $$x^{2} = 1$$, take the square root of both sides to get $$x = \pm\sqrt{1}$$ which simplifies to $$x = \pm 1$$

Step 4: Combine the results

The equation $$(x+3)\left(1-x^{2}\right)=0$$ has three solutions: 1. $$x = -3$$ 2. $$x = 1$$ 3. $$x = -1$$

Key Concepts

Factoring

When we talk about factoring in the context of algebra, we're referring to the process of breaking down an equation into simpler parts called factors. These factors, when multiplied together, will give you the original equation. In our problem, the given equation \[(x+3)(1-x^{2})=0\]is already factored for us, which means it is already in its simplest form as a product of two factors:

• \((x + 3)\)
• \((1 - x^{2})\)

This makes it easier to solve, as we can deal with each factor separately. Factoring is especially useful for quadratic equations, where we look to express the quadratic as a product of two binomials. Once factored, each part can be set to zero to find the solution or roots of the equation.

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that states: if a product of two numbers is zero, then at least one of the numbers must be zero. This property is especially helpful when solving factored equations like ours. We start with the equation \[(x+3)(1-x^{2}) = 0\] Using the zero product property, we know:

• Either \((x+3) = 0\)
• Or \((1-x^{2})=0\)

By applying this property, we can separate the equation into simpler linear equations:

• \(x + 3 = 0\)
• \(1 - x^{2} = 0\)

Solving these gives us the roots of the original equation. This property is indispensable in algebra as it allows us to easily solve polynomials that are zero.

Quadratic Roots

Once we have used factoring and the zero product property, finding the roots of a quadratic becomes straightforward. A root (or solution) of an equation is any value that makes the equation true. In our example, setting our factored components to zero gives us two types of equations to solve:For the factor \((x + 3)\):- Solving \(x + 3 = 0\) gives us \(x = -3\) For the factor \((1 - x^{2})\):- Solving \(1 - x^{2} = 0\), we rearrange to find \(x^{2} = 1\) - The solutions are \(x = 1\) and \(x = -1\), since any number squared is positive, there are two solutions.Hence, the roots of the quadratic equation \((x+3)(1-x^{2})=0\) are \(x = -3\), \(x = 1\), and \(x = -1\). Understanding the concept of quadratic roots is essential for solving many kinds of algebraic equations.