How do you factor X3 Y3?

How do you factor X3 Y3?

Factorization of x3 + y 3

1. It can be seen in most book that x3 + y3 can be factorized by dividing the expression by (x + y). After division we get a quotient of (x2 – xy + y2) with no remainder.
2. However, this method involves knowing the factor (x + y) beforehand (and the understanding of Factor Theorem).

What is a factor in math?

Factor, in mathematics, a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder. For example, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly. The other factors of 12 are 1, 2, 4, and 12.

What is the importance of factor theorem?

We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

Who invented the factor theorem?

The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao. The Chinese remainder theorem addresses the following type of problem.

What is factor theorem in determinants?

If each element of a matrix A is a polynomial in x and if | A | vanishes for x = a, then (x – a) is a factor of | A |. (ii) If we substitute b for a in the determinant | A |, any two of its rows or columns become identical, then | A | = 0, and hence by factor theorem (a – b) is a factor of | A |.

How do you find factors of determinants?

If each element of a matrix A is a polynomial in x and if |A| vanishes for x = a, then (x – a) is a factor of | A |. (i) This theorem is very much useful when we have to obtain the value of the determinant in ‘factors’ form.

What is the property of determinant?

Determinant evaluated across any row or column is same. If all the elements of a row (or column) are zeros, then the value of the determinant is zero.

What is cyclic polynomial?

Cyclic polynomials are polynomial functions that are invariant under cyclic permutation of the arguments. These polynomials are closely related to symmetric polynomials as all symmetric polynomials are cyclic (but not vice versa).

What is a cyclic expression?

Cyclic Expressions : Some expressions involing 3 variables, say a, b and c, remains same, even if a is replaced by b, b is replaced by c and c is replaced by a simultaneously. Such expression is called cyclic expressions. If (a+b) is a factor of a cyclic expression, then (b+c) and (c+a) are also factors of it.

What is a cyclic function?

Definition of cyclic function. : a mathematical function that changes in value by an additive constant whenever its variable arguments pass continuously through a cycle of values.

What is symmetric expression?

A symmetric polynomial is a polynomial where if you switch any pair of variables, it remains the same. For example, x 2 + y 2 + z 2 x^2+y^2+z^2 x2+y2+z2 is a symmetric polynomial, since switching any pair, say x and y, the resulting polynomial y 2 + x 2 + z 2 y^2+x^2+z^2 y2+x2+z2 is the same as the initial polynomial.

What is meant by periodic function?

A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians. Any function that is not periodic is called aperiodic.

What other situations might be modeled by a periodic function?

Other Models of Periodic Behavior For example, high tides and low tides can be modeled and predicted using periodic functions because scientists can determine the height of the water at different times of the day (when the water level is low, the tide is low).

What is cyclic process in chemistry?

In a cyclic process, the system starts and returns to the same thermodynamic state. A cyclic process is the underlying principle for an engine. If the cycle goes counterclockwise, work is done on the system every cycle. An example of such a system is a refrigerator or air conditioner.

What is cycle process?

The process in which the initial and final state is the same is known as a cyclic process. It is a sequence of processes that leave the system in the same state in which it started. If the cycle goes clockwise, the system does work. If the cycle goes anticlockwise, then the work is done on the system every cycle.

What is Isochoric process?

An isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant.

What is true cyclic process?

The cyclic process is the process which has the same initial and the final state of the system. So, the energy of the system in its initial and final position is the same. So, the work done by the system is equal to the heat supplied to the system. Option C is correct.

What is zero in a cyclic process?

The change in energy in a cyclic process is zero, since the initial and final states are the same. The work done and the quantity of heat gained in such a process are therefore the same with opposite signs (R = –Q).

Which of the following is not possible in cyclic process?

Solution : Work dine on the system is always taken as negative.

Which of the following is incorrect for cyclic process?

For a cyclic process work done is not equal to zero, because work is not a state function. It depends on the path. While ΔH,ΔU,ΔS are state function. Was this answer helpful?

What is the change in internal energy in a cyclic process?

The net change in internal energy is zero since the system returns to the same thermodynamic state (the definition of a cyclic process) and internal energy is a property and therefore only a function of the state of the system. So for a cyclic process, Q=W.

Which of the following statement is true for a cyclic process?

For cyclic process W=−q. This is because for cyclic process ΔU=0 and ΔU=W+q.

Which of the following is not a state function?

Solution : Work is not a state function as during a process its value depends on the path followed. The value of enthalpy, internal energy entropy depends on the state and not on the path followed to get that state, hence these are state functions.