You can't "solve" an expression. You can, however, solve an equation or inequality. Is the equation y^4 + 4y - 69 = 0?phillyolly said:Homework Statement
Solving y^4+4y-69
I don't know of any ways to solve quartic (fourth-degree) equations analytically, but there might be some. There are ways to solve cubics (third-degree), but even they are fairly involved to solve.phillyolly said:, I got the following:
By manually plugging in, I found that y= -3 and y=2.76.
However, I would like to ask you if there are other good, efficient ways to find solutions?
(Without using sophisticated math tools. Imagine, I have this prob on the exam)
Thank you!
One approach is to use factoring. First, try to find any common factors between the terms. In this case, there are no common factors. Then, try to factor the expression by grouping. For example, we can group the first two terms together and the last two terms together: (y^4+4y) - (69). This can be rewritten as y(y^3+4) - 69. Since there is no common factor between y and (y^3+4), we can't factor further. Therefore, the expression cannot be simplified without using advanced math tools.
No, the quadratic formula can only be used to solve equations in the form of ax^2+bx+c=0, where a, b, and c are constants. Our equation, y^4+4y-69, is not in this form and therefore cannot be solved using the quadratic formula.
No, substitution is usually used to solve systems of equations, where there are two or more equations with multiple variables. In our equation, there is only one variable (y) and no other equations to substitute it into. Therefore, substitution cannot be used to solve this equation.
No, logarithms are used to solve exponential equations, where the variable is in the exponent. In our equation, the variable is not in the exponent and therefore logarithms cannot be used to solve it.
Yes, another approach is to use the rational root theorem. This theorem states that if a polynomial has rational roots, they must be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In our equation, the constant term is -69 and the leading coefficient is 1. Therefore, the only possible rational roots are ±1, ±3, ±23, and ±69. By substituting these values into the equation, we can determine if any of them are roots and then use long division to factor the equation and find the remaining roots. However, this method can become tedious and may still require some basic understanding of polynomial functions.