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Cubic and quartic equations can be solved by similar formulas, and this has been known since the 16th century: del Ferro, Cardan, and Tartaglia are all credited with having discovered the cubic equation, and Ferrari is credited with having discovered the quadratic equation. Higher-degree equations make the question more complicated. Similar formulas can be used to solve cubic and quadratic problems. (It is interesting to note that Ramanujan, who was largely self-taught and possesses what is often considered to be the most brilliant mathematical intellect of the 20th century, came up with his own approach for solving the quadratic after he was shown how to solve the cubic.) The outlook is even more gloomy for higher-degree equations: Abel demonstrated in the first half of the 19th century that equations of fifth degree and higher do not have formulas that are similar to those found in lower-degree equations. Even though Evariste Galois did not survive to see it accomplished, the same result was a natural consequence of the work that he undertook. Galois was a French mathematician who passed away in a duel before he became 21 years old.
What are Higher Degree Equations:
In mathematics, the degree of a polynomial is defined as the maximum degree of any of the monomials (individual terms) that the polynomial contains that have coefficients that are not zero. The degree of a term is an integer that cannot be less than zero because it is calculated as the sum of the exponents of the variables that appear in the term.
Solving Higher Degree Equations:
We can utilise substitution to convert a given equation into a quadratic equation, then solve the quadratic equation to find the solutions to the original equation.
For example, suppose we have the equation:
ax4 + bx² + c = 0
If we let z = x2, then substitute this into the original equation, we can rewrite it as:
a(x²)² + b(x²) + c = 0
az² + bz + c = 0,
solvable quadratic equation (by factoring or using the quadratic equation).
Then after solving, we can set the solutions for z equal to x², then solve for x.
Importance of Higher Degree Equations:
Higher-Degree Equations are utilised not only in engineering and other math- and computer-based fields, but also in management, commerce, and even farming. Variables and constants are used to build expressions specifying values that are known and unknown. Knowledge of polynomials is required for a wide variety of vocations, including those in the medical, legal, and financial fields.
Practice Problem:
- Find a polynomial expression for the function x = 0, 3, and –1.
Solution: You can find the polynomial if you know all the zeros, like we mentioned earlier. A factor (x – c) in the polynomial corresponds to a zero at x = c. Let’s use this fact to construct the polynomial p(x) that corresponds to the zeros given in the problem.
p(x) = (x – 0)(x – 3)(x – (–1)) = x(x – 3)(x + 1)
Now let’s discover the polynomial expression.
p(x) = x(x² – 2x – 3) = x3 – 2x² – 3x
Graphing, testing each zero, or both can corroborate the result. Let’s just test each zero.
p(0) = (0)³ – 2.(0)² – 3.(0) = 0
p(3) = (3)³ – 2.(3)² – 3.(3) = 27 – 2.(9) – 9 = 27 – 18 – 9 = 0
p(–1) = (–1)³ – 2.(–1)² – 3.(–1) = –1 – 2 + 3 = 0
At each of the associated x values, this function equals zero.
Conclusion:
After demonstrating how any higher-order equation can be expressed as a system of first-order equations, the authors explore the existence and uniqueness criteria. Next, they extend the first-order numerical approximation methods to systems. The concept of phase portraiture is presented for planar systems. An overview of vectors and matrices is provided prior to a more in-depth discussion of the systematic presentation of planar systems of autonomous linear equations. The eigenvalues of the matrix of coefficients are used to provide a comprehensive description of the stability of homogeneous systems.
