As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

Since p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

In more general sense, each column is treated as a polynomial over GF ( 2 < sup > 8 </ sup >) and is then multiplied modulo x < sup > 4 </ sup >+ 1 with a fixed polynomial c ( x ) = 0x03 · x < sup > 3 </ sup > + x < sup > 2 </ sup > + x + 0x02.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

These points form a line, and y = x is said to be the equation for this line.

This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).

The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

In the < tt > MixColumns </ tt > step, each column of the state is multiplied with a fixed polynomial c ( x ).

Head of a Faun ( c. 1595 ) 181 x 187 mm Pen and brown ink on laid paper National Gallery of Art, Washington.

The slope field of F ( x ) = ( x < sup > 3 </ sup >/ 3 )-( x < sup > 2 </ sup >/ 2 )- x + c, showing three of the infinitely many solutions that can be produced by varying the Constant of integration | arbitrary constant C.

The horizontal line y = c is a horizontal asymptote of the function y = ƒ ( x ) if

In the first case, ƒ ( x ) has y = c as asymptote when x tends to −∞, and in the second that ƒ ( x ) has y = c as an asymptote as x tends to +∞

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

The coefficient a in the term of x < sup > b </ sup > y < sup > c </ sup > is known as the binomial coefficient or ( the two have the same value ).

If the weight of the roadway per unit length is w and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable from c to r is wx where x is the horizontal distance between c to r. Proceeding as before gives the differential equation

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

Sum_sqr + x * x

sum1 + x

In plain text, and in the TeX mark-up language, the caret symbol "^" represents exponents, so is written as " x ^ 2 ".

x ^ n =

"". join ( chr ( x ^ 0x36 ) for x in xrange ( 256 ))

:< math > p ( x ) = b_1x ^

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

solve ( x ^ 2 = 1 );

a * x ^ 2 + b * x + c = 0 ;

a * x ^ 3 + b * x ^ 2 + c * x + d = 0 ;

a * x ^ 4 + b * x ^ 3 + c * x ^ 2 + d * x + e = 0 ;

Some adaptations of the Latin alphabet are augmented with ligatures, such as æ in Old English and Icelandic and Ȣ in Algonquian ; by borrowings from other alphabets, such as the thorn þ in Old English and Icelandic, which came from the Futhark runes ; and by modifying existing letters, such as the eth ð of Old English and Icelandic, which is a modified d. Other alphabets only use a subset of the Latin alphabet, such as Hawaiian, and Italian, which uses the letters j, k, x, y and w only in foreign words.

It is the coefficient of the x < sup > k </ sup > term in the polynomial expansion of the binomial power ( 1 + x )< sup > n </ sup >.

The coefficient of x < sup > n − k </ sup > y < sup > k </ sup > is given by the formula

E has two clauses ( denoted by parentheses ), four variables ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >, x < sub > 4 </ sub >), and k = 3 ( three literals per clause ).

However, as 1 + ( k + 1 ) x + kx < sup > 2 </ sup > ≥ 1 + ( k + 1 ) x ( since kx < sup > 2 </ sup > ≥ 0 ), it follows that ( 1 + x )< sup > k + 1 </ sup > ≥ 1 + ( k + 1 ) x, which means the statement is true for r = k + 1 as required.

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

When the boot loader detected a CP / M floppy, the Aster would reconfigure its internal memory architecture on the fly to optimally support CP / M with 60 KB free RAM for programs ( TPA ) and an 80 x 25 display.

Changing x to x / b stretches the graph horizontally by a factor of b. ( think of the x as being dilated )

The Battle of Alexander at Issus | The Battle of Issus / Alexander, 1529, Wood, 158, 4 x 120, 3 cm Alte Pinakothek, Munich.

Consider the graph of the equation y = 1 / x shown to the right.

The coordinates of the points on the curve are of the form ( x, 1 / x ) where x is a number other than 0.

But no matter how large x becomes, its reciprocal 1 / x is never 0, so the curve never actually touches the x-axis.

An example is ƒ ( x ) = x − 1 / x, which has the oblique asymptote y = x ( m = 1, n = 0 ) as seen in the limits