Many fourth-degree polynomial equations can be transformed into quadratic equations using a well-known high school trick of substitution. This also works perfectly for polynomials in complex numbers.
See how to turn these equations into lower-degree equations.
When given a limit to calculate that involves some subtraction with a root (and which obviously couldn’t be calculated more simply), such as: “Something – square root of something”, “square root of something – Something”, or “square root of something – square root of something”, we used a trick I call – “multiplication by the conjugate.”
We simply multiplied this expression by its counterpart with a plus sign, or rather by a fraction where this counterpart was in the numerator and the denominator.
What to do when the roots are cube roots?
In the limits of sequences, sometimes the expression contains the sum of squares or cubes of consecutive natural numbers. What then?
The answer is simple: formulas for the sum of squares and the sum of cubes of consecutive natural numbers. They go like this…
I’ve added a new article with a video to the “Lectures” section (on the right sidebar): Cyclometric Functions.
It’s dedicated to cyclometric functions (arcsinx, arccosx, arctgx, arcctgx) and is divided into two parts. In the first part, I quickly show how to calculate them, and in the second part, I delve deeper into the topic (definitions, graphs).
In previous posts, I demonstrated how to use Euler substitutions in integrals with the square root of the polynomial ax^2+bx+c.
Euler substitutions of the first kind were used when a>0, and second kind Euler substitutions when c>0. In this post, we will deal with the third and final kind of Euler substitutions that can be used when the quadratic polynomial in the integral has TWO DISTINCT roots x1, x2, meaning its discriminant is positive. See what to do in this case.
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