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For example, let f(x) = 6x 4 − 2x 3 + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x 4, −2x 3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x ...
Simplifying this further gives us the solution x = −3. It is easily checked that none of the zeros of x (x + 1)(x + 2) – namely x = 0, x = −1, and x = −2 – is a solution of the final equation, so no spurious solutions were introduced.
Substitutions of hyperbolic functions can also be used to simplify integrals. [ 4 ] For example, to integrate 1 / a 2 + x 2 {\displaystyle 1/{\sqrt {a^{2}+x^{2}}}} , introduce the substitution x = a sinh u {\displaystyle x=a\sinh {u}} (and hence d x = a cosh u d u {\displaystyle dx=a\cosh u\,du} ), then use the identity cosh 2 ( x ...
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
If we solve this equation, we find that x = 2. More generally, we find that + + + + is the positive real root of the equation x 3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic ratio ρ, approximately equal to 1.3247.
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
Our country's tax code spans 4 million words, and its complexity contributes to why a fifth of eligible Americans don't claim one of the most valuable tax credits available.
In mathematics, like terms are summands in a sum that differ only by a numerical factor. [1] Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients.