Question

Find all real solutions of the polynomial equation. $$x^{5}-7 x^{4}+10 x^{3}+14 x^{2}-24 x=0$$

Short answer

So, the only real solution for the equation \(x^{5}-7 x^{4}+10 x^{3}+14 x^{2}-24 x=0\) is \(x=0\).

Step-by-step solution

Factorize the Polynomial

Factor out the common factor which is x in this case.\(x (x^{4}-7 x^{3}+10 x^{2}+14 x-24)=0\)

Simplify the Equation

After factoring out x, the polynomial should be simplified which will result in two factors \(x=0\) and \(x^{4}-7 x^{3}+10 x^{2}+14 x-24=0\)

Solve Equation

By applying the Null Factor Law - because two quantities multiplied together equal zero, then at least one of them must be zero, therefore, x=0. Now we must find the roots of the other quadratic equation by trying to factorize it or use other methods like Rational Root Theorem or Synthetic Division.

Use Rational Root Theorem

This theorem gives us a list of potential rational roots of a polynomial equation. After using it, it comes out that equation \(x^{4}-7 x^{3}+10 x^{2}+14 x-24=0\) does not have any rational roots.

Use Synthetic Division

Using synthetic division will allow us to determine the real roots of the equation \(x^{4}-7 x^{3}+10 x^{2}+14 x-24=0\). Unfortunately, this method also indicates that there are no other real roots.

Key concepts

Factorization

Factorization is a crucial method in solving polynomial equations by breaking them down into simpler, easily solvable pieces. Imagine you have an equation like \(x^5 - 7x^4 + 10x^3 + 14x^2 - 24x = 0\). The first step is to check for a common factor among all the terms. Here, we found that each term contains at least one \(x\), so \(x\) becomes our common factor.
By factoring out \(x\), the polynomial simplifies to \(x (x^4 - 7x^3 + 10x^2 + 14x - 24) = 0\). This simplification shows that one possible solution is \(x = 0\). This process is based on the idea that in a multiplication of terms, if any term is zero, the complete expression equals zero.
Once factorization is complete, other factors can be further resolved, if possible, to uncover additional roots. However, as seen in our exercise, some polynomials require more than factorization for complete solving.

Rational Root Theorem

The Rational Root Theorem is a powerful tool in polynomial algebra. It helps us predict potential rational roots of a given polynomial. For the equation \(x^4 - 7x^3 + 10x^2 + 14x - 24 = 0\), this theorem offers possible rational roots by using the factors of the constant term (here \(-24\)) and the leading coefficient (which is \(1\)).
We list all combinations of these factors, both positive and negative, as potential roots. This might seem extensive, but it helps to quickly identify which, if any, rational numbers are roots.
Even though in our example, this method proved that \(x^4 - 7x^3 + 10x^2 + 14x - 24 = 0\) does not have rational roots, the theorem remains a useful initial test when tackling complicated polynomial equations. It saves time by ruling out options that do not satisfy the equation.

Synthetic Division

Synthetic Division is an efficient method of dividing polynomials, especially when applying it to find roots. Unlike traditional long division, synthetic division is quicker and involves dealing with coefficients only. For our polynomial \(x^4 - 7x^3 + 10x^2 + 14x - 24 = 0\), synthetic division was used after ruling out rational roots.
The process requires choosing a potential root, usually from what Rational Root Theorem suggests, and testing it. This involves simple arithmetic operations where you write down coefficients and then divide step by step.

• Write down the coefficients.
• Use your test root in a quick calculation.
• Observe if the remainder is zero; if yes, the test root is indeed a real root.

Synthetic division revealed no other real roots for our polynomial, confirming the solution found via factorization. It's a great technique to finalize and verify roots in polynomial equations.