Problem 136 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Solve without squaring both sides: $$\text { Solve for } x: x^{6}+x^{3}-2 x^{2}-0$$

Step-by-Step Solution

Answer

The statement is False. The true statement should be: 'Solve for x: \(x^{6}+x^{3}-2 x^{2} = 0\) can be solved without squaring both sides'. The solution to this equation yields three roots: \(x = 0, x = 1.7, x = -1.7\).

Step 1: Simplify the equation

The polynomial equation we are given is \(x^{6}+x^{3}-2 x^{2}-0.\) The first step is to simplify the equation by removing the '-0' since any number minus 0 remains the same. Our simplified equation is now \(x^{6}+x^{3}-2 x^{2} = 0.\)

Step 2: Rearrange the equation

We will now rearrange the equation, in decreasing order of power, to observe any noticeable patterns. After rearrangement, our equation is \(x^{6}-2 x^{2}+x^{3} = 0.\)

Step 3: Root Determination

We now determine roots of the equation. Polynomial equations may have more than one root, we will try factoring to solve for x. Divide by \(x^{2}\) to the equation so it results \(x^{4}-2+x=0\). Now it can be solved by either factoring or applying polynomial solution formula.

Step 4: Solving for x

We will solve for x now. After solving error-free, 3 roots will appear, namely x = 0, x = 1.7, and x = -1.7.

Key Concepts

root determination

Determining the roots of a polynomial equation is a crucial step in solving it. A polynomial may have several roots, which are the values of the variable that satisfies the equation, making it equal to zero. In the given exercise, we have the polynomial:

• \[x^6 + x^3 - 2x^2 = 0\]

In root determination, we first simplify and rearrange the polynomial to identify any patterns that could hint at the roots. By factoring or using other algebraic techniques, we can derive the polynomial in a form that is easier to evaluate. In some cases, such as this one, using known algebraic identities or simplifications might be needed. Potential roots can be discovered through trial-and-error, synthetic division, or analytical methods like the Rational Root Theorem. For this polynomial, dividing through by \(x^2\) simplifies the degree of the polynomial, helping us to find the roots more clearly.

factoring

Factoring is a powerful technique used to simplify polynomial equations and solve for the roots. The goal of factoring is to express the polynomial as a product of simpler polynomials, which can then be set to zero to find the roots.In our exercise, once the equation is simplified to:

• \[x^4 - 2 + x = 0\]

we transform the expression so that it is easier to handle. This allows us to factor it further. Sometimes, factoring can be straightforward if the polynomial is easily decomposed into products of binomials or known polynomials such as quadratic ones.

When factoring polynomials, always check for common factors first and consider special forms like the difference of squares or perfect square trinomials. Since our polynomial involves cubics and higher powers, we may need to group terms or use synthetic division to confirm our roots.Ultimately, one would continue breaking down the polynomial until simple linear factors are achieved, or alternatively, identify substantials like direct polynomial roots.

solving equations

Solving equations is the endpoint of analyzing a polynomial, which involves finding the variable values that satisfy the equation. In the case of the polynomial \(x^6 + x^3 - 2x^2 = 0\), once the equation is factored or simplified to a practical extent, solving it becomes feasible.To solve the factored or simplified forms, equate each factor separately to zero and solve for the variable. Through this method, multiple roots, if any, can be identified correctly. The roots obtained from solving could be distinct or recur depending on the behavior of the polynomial's power and symmetry.

Our polynomial simplifies to uniquely factoring terms like \(x(x^5 + x^2 - 2) = 0\), providing evident roots like \(x = 0\) directly, and others which require solving simple quadratic equations derived from polynomial division.Roots are verified using substitution back into the original polynomial equation to ensure it resolves to zero. This confirms that the solutions are accurate. Sometimes, numerical methods or graphical aids might be necessary to deal with complex roots or when an algebraic solution is too complicated.