Posted by Anonymous on Monday, August 17, 2009 at 2:04am.
If the question is concerned with a particular term of a polynomial, a coefficient is the numeric or constant (or a combination) part of the term that precedes the variable(s). The exponents are the powers to which the variables are raised.
For example, the first term of the polynomial
5x² + 3x + 1
is 5x².
5 is the coefficient, 2 is the exponent of the variable x.
Thus the operations that we can associate with coefficients are mostly addition and subtraction in the grouping of like terms, such as in simplifying
5x² + 3x + 1 + 4x² + 2x + 3
=(5+4)x² + (3+2)x + (1+3)
= 9² + 5x + 4
Since like terms have variables raised to the same power (exponent), no operations are done on the exponents when grouping like terms.
On the other hand, during a multiplication or division of a polynomial by a monomial (single term) or polynomial, both the coefficients and the variables are multiplied together. For example,
5x² * 4x³
will be calculated as
5x² * 4x³
= (5*4)* x²*x³
= 20 x²*sup3;
= 20 x^{2+3}
= 20x^{5}
Here the coefficients are multiplied together and the exponents are added because of the laws of exponents.
In the less common cases where a term is raised to a power, the coefficient is raised to the same power, while the exponent of the variables are simply multiplied according to the laws of exponents.
For example,
(5x²)²
=(5²)*(x²)²
= 25 * (x^{2*2}
= 25 x^{4}
In short, while both the coefficients and the exponents are subject to the four operations of add, subtract, multiply and divide, the coefficient could be raised to a power in certain cases.