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When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.
Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
In case I, the exponent 5 does not divide the product xyz. In case II, 5 does divide xyz. Case I for n = 5 can be proven immediately by Sophie Germain's theorem(1823) if the auxiliary prime θ = 11. Case II is divided into the two cases (cases II(i) and II(ii)) by Dirichlet in 1825. Case II(i) is the case which one of x, y, z is
In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form , with q and p being coprime integers. The rational root test allows finding q and p by examining a finite number of cases (because q must be a divisor of a , and p must be a divisor of d ).
[5] Exponential functions with bases 2 and 1/2 The base of an exponential function is the base of the exponentiation that appears in it when written as x → a b x {\displaystyle x\to ab^{x}} , namely b {\displaystyle b} . [ 6 ]
The degree is the sum of the exponents on the variables; in this example, = + + A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5.
Thus 350 is written as 3.5 × 10 2. This form allows easy comparison of numbers: numbers with bigger exponents are (due to the normalization) larger than those with smaller exponents, and subtraction of exponents gives an estimate of the number of orders of magnitude separating the numbers.
The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution.
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