- #1
navee
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How can we factor a question which is too the power of 4 or quartic. for example ax^4+bx^3+cx^2+dx+e, but a has a grreater value than 1.
Solving Quartic Equations: A > 1
- Thread starter navee
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In summary, factoring a fourth order polynomial equation with a leading coefficient greater than 1 can be solved using the rational zeros test and synthetic division to reduce the equation to a quadratic. Alternatively, numerical methods such as Newton's Method can be used. In general, factoring higher order polynomials is a complex process and can be solved using complex numbers or by finding the solutions of the equation. There is no general method for factoring, but the solutions can be used to obtain the factors of the polynomial equation.
- #2
shaner-baner
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I'm not sure, but I think that fourth order equations are solvable with some weird equation like third order. In practice, you might start with the rational zeros test. What you do is: take all possible factors of 'e' and divide by all possible factors of 'a'. Then test these to see if they are zeros (if q is a zero, x-q is a factor). Using a procedure called synthetic division let's you use the factors that you find to reduce the order of the equation. When the equation has finally been reduced down to a quadratic, then you can use the quadratic formula. Alternately, there are 'numerical' methods that can find approximations of the zeros, (i.e. Newtons Method). There are a few other tricks. For example. If it looks like ax^4+bx^2+c, you can just use the quadratic formula to solve for x^2, and then take the square root of your answers.
In general factoring higher order polynomials is a tricky business. Hope I answered your question
In general factoring higher order polynomials is a tricky business. Hope I answered your question
- #3
HallsofIvy
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Theoretically, any polynomial equation can be factored completely into linear factors if you use complex numbers, linear and quadratic factors if you stay in real numbers.
Yes, there exist a "quartic" formula. Here is a website with a (comparatively) simple explanation: http://web.usna.navy.mil/~wdj/book/node95.html
There is, however, no general method for factoring except by solving the equation and then using the solutions to get the factors: If a, b, c are solutions, the we can factor as p(x-a)(x-b)(x-c).
Yes, there exist a "quartic" formula. Here is a website with a (comparatively) simple explanation: http://web.usna.navy.mil/~wdj/book/node95.html
There is, however, no general method for factoring except by solving the equation and then using the solutions to get the factors: If a, b, c are solutions, the we can factor as p(x-a)(x-b)(x-c).
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FAQ: Solving Quartic Equations: A > 1
How do you solve a quartic equation with a coefficient greater than 1?
To solve a quartic equation with a coefficient greater than 1, you can use the quartic formula, which involves finding the roots of a complex number. This formula is more complicated than the quadratic formula and may require the use of a calculator or computer program to solve.
Can a quartic equation with a coefficient greater than 1 have multiple solutions?
Yes, a quartic equation with a coefficient greater than 1 can have up to four solutions, depending on the specific values of the coefficients and constants in the equation. These solutions may be real or complex numbers.
What is the difference between solving a quadratic equation and a quartic equation with a coefficient greater than 1?
The main difference between solving a quadratic equation and a quartic equation with a coefficient greater than 1 is the complexity of the formulas used. Quadratic equations can be solved using the quadratic formula, while quartic equations require the use of the quartic formula, which is more complicated and may require the use of technology.
Are there any shortcuts or tricks for solving quartic equations with a coefficient greater than 1?
There are some techniques that can be used to simplify solving quartic equations with a coefficient greater than 1, such as factoring and using the rational root theorem to identify potential roots. However, these methods may not always be applicable and the use of the quartic formula may still be necessary.
Are quartic equations with a coefficient greater than 1 commonly used in real-world applications?
Quartic equations with a coefficient greater than 1 are less common in real-world applications compared to quadratic equations, but they can still be found in fields such as physics, engineering, and economics. They may also be used to model complex systems or in advanced mathematical problem-solving.