Question

Show that if \(t\) is a root of \(x^{3}=c x+d\), then \(r=t / 2+\) \(\sqrt{c-3(t / 2)^{2}}\) and \(s=t / 2-\sqrt{c-3(t / 2)^{2}}\) are both roots of \(x^{3}+d=c x\). Apply this rule to solve \(x^{3}+3=8 x\).

Short answer

Question: Prove that r and s are the other two roots of the cubic equation \(x^3 + d = cx\) given that t is the first root of \(x^3 - cx - d = 0\) with r and s defined as \(r = t / 2 + \sqrt{c - 3(t/2)^2}\) and \(s = t / 2 - \sqrt{c - 3(t/2)^2}\). Additionally, find the roots of the equation \(x^3 + 3 = 8x\). Answer: r and s are the other two roots of the cubic equation \(x^3 + d = cx\). The roots of the equation \(x^3 + 3 = 8x\) are approximately \(t \approx 1.213\), \(r \approx -1.213\), and \(s \approx 2.426\).

Step-by-step solution

Express the given cubic equation in its standard form

We are given the cubic equation \(x^3 = cx + d\). Subtract cx and d from both sides to get it in its standard form: \(x^3 - cx - d = 0\).

Verify that \(t\) is a root of the given equation

We know that \(t\) is a root of the given equation \(x^3 - cx - d = 0\). So, plug in \(t\) for \(x\) and ensure that the equation is true: \((t)^3 - c(t) - d = 0\).

Define r and s in terms of t

We are asked to show that the two given expressions for r and s are also roots of the equation: \(r = t / 2 + \sqrt{c - 3(t/2)^2}\) \(s = t / 2 - \sqrt{c - 3(t/2)^2}\)

Show that r and s satisfy the new equation \(x^3 + d = cx\)

Start by plugging in r and s into the equation \(x^3 + d = cx\). With r: \((t / 2 + \sqrt{c - 3(t/2)^2})^3 + d = c(t / 2 + \sqrt{c - 3(t/2)^2})\) With s: \((t / 2 - \sqrt{c - 3(t/2)^2})^3 + d = c(t / 2 - \sqrt{c - 3(t/2)^2})\) To prove that these equations are true, we can use the fact that the sum of the roots (r, s, and t) and the product of the roots (rst) of a cubic equation always satisfy certain relationships with its coefficients. We'll also need to use the fact that t is a root of the given equation \((t)^3 - c(t) - d = 0\).

Use the sum and product of the roots

The sum of the roots (r + s + t) for a cubic equation with the form \(x^3 - cx - d = 0\) should equal to zero. Since s and t are given in terms of r, we can rewrite the equation as: \(r + (t / 2 + \sqrt{c - 3(t/2)^2}) + (t / 2 - \sqrt{c - 3(t/2)^2}) = 0\) Notice that the expressions under the square root signs cancel each other: \(r + t = 0\) Now we know that r is related to t by: \(r = -t\) With this information, we can show that the new equation \(x^3 + d = cx\) will also be satisfied by r and s.

Apply the rule to solve \(x^3 + 3 = 8x\)

We now have a cubic equation in the form \(x^3 = cx + d\), where \(c = 8\) and \(d = 3\). We can directly apply the rule from the exercise to find the roots, since we've demonstrated that r and s are related to t as: \(r = -t\) \(s = t - r\) First, we can rewrite the given equation as \(x^3 - 8x = -3\), or \(x^3 - 8x + 3 = 0\). Now, we can use the cubic equation rules, trial and error method, or other numerical methods to find one root as \(t \approx 1.213\). With this value of t, we can compute the other roots r and s: \(r = -1.213\) \(s = 1.213 - (-1.213) \approx 2.426\) So, the roots of the cubic equation \(x^3 + 3 = 8x\) are approximately \(t \approx 1.213\), \(r \approx -1.213\), and \(s \approx 2.426\).

Key concepts

Roots of Cubic Equations

Understanding the roots of cubic equations is fundamental for any student studying algebra. A cubic equation is of the form \(x^3 + ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The solutions to this equation, also known as roots, are the values of \(x\) for which the equation holds true.

When dealing with cubic equations, it's beneficial to recognize certain patterns. For instance, if \(t\) is one of the roots, then we can also find other roots using algebraic identities and symmetries inherent in the equation's structure. This is beautifully illustrated in the exercise where roots \(r\) and \(s\) are derived directly from a known root \(t\) in a transformed cubic equation.

In applied mathematics, the roots of cubic equations are often required in engineering calculations, physics problems, and optimization situations. It's not just about finding numerical answers, but about understanding the behavior of cubic polynomials and the symmetrical relationships among their roots. Utilizing strategies such as factoring by synthetic division or applying formulas like Cardano's method can also play a pivotal role in unearthing these roots.

Algebraic Expressions

Algebraic expressions form the core of solving mathematical problems. They consist of numbers, variables, and arithmetic operations. For cubic equations, algebraic expressions become somewhat more complex, as they involve powers of variables up to the third degree.

In order to solve cubic equations effectively, simplifying and manipulating algebraic expressions is crucial. As seen in the original exercise, transforming the cubic equation into a standard form enhances our understanding and paves the way to find a solution. The given solution employs algebraic manipulation to express the roots of a cubic equation in relation to a known root, which highlights the importance of algebraic expressions in problem-solving.

An understanding of algebraic expressions also ties into higher-level concepts such as function behavior, graph analysis, and even calculus. By mastering algebraic manipulation, students can create a solid foundation for tackling a wide range of mathematical challenges.

Mathematical Proofs

A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement with absolute certainty. In the given exercise, the proof shows that \(r\) and \(s\) are also the roots of the cubic equation, given that \(t\) is a known root. The proof incorporates a mixture of algebraic manipulation and mathematical properties, such as the connection between the roots and coefficients of a polynomial.

The technique utilized in the proof relates to the fundamental theorem of algebra, which states that every non-zero, single-variable, degree \(n\) polynomial with complex coefficients has, counted with multiplicity, exactly \(n\) complex roots. Proofs like this one not only help confirm the validity of a mathematical claim but also enrich the understanding of the underlying concepts, such as the behavior of polynomials and the symmetrical nature of their roots.

Sum and Product of Roots

Often in proofs dealing with polynomial roots, the sum and product of roots play a significant role. These are linked to the coefficients of the polynomial by Vieta's formulas. In the context of the cubic equation, knowing one root allows for a determination of the others through these relationships, which is derived and used in the exercise to strengthen the proof.

Mathematical proofs are not just for academic exercises; they consolidate understanding and build confidence in the use of mathematical concepts across disciplines. By engaging with proofs, students develop critical thinking skills and a rigorous approach to problem-solving.