Monomials and polynomials

Monomial. Numerical factor. Coefficient. Similar (like) monomials.
Addition of monomials. Reducing of like terms. Multiplication and
division of monomials. Polynomial. Degree of polynomial.
Multiplication of sums and polynomials.

Monomial is a product of two or some factors, each of them is either a number, or a letter, or a power of a letter. For example,

3 a ² b ⁴ , b d ³ , – 17 a b c

are monomials. A single number or a single letter may be also considered as a monomial. Any factor of a monomial may be called a coefficient. Often only a numerical factor is called a coefficient. Monomials are called similar or like ones, if they are identical or differed only by coefficients. Therefore, if two or some monomials have identical letters or their powers, they are also similar (like) ones. Degree of monomial is a sum of exponents of the powers of all its letters.

Addition of monomials. If among a sum of monomials there are similar ones, he sum can be reduced to the more simple form:

a x ³ y ² – 5 b ³ x ³ y ² + c ⁵ x ³ y ² = ( a – 5 b ³ + c ⁵ ) x ³ y ² .

This operation is called reducing of like terms. Operation, done here, is called also taking out of brackets .

Multiplication of monomials. A product of some monomials can be simplified, only if it has powers of the same letters or numerical coefficients. In this case exponents of the powers are added and numerical coefficients are multiplied.
E x a m p l e :

5 a x ³ z ⁸ ( – 7 a ³ x ³ y ² ) = – 35 a ⁴ x ⁶ y ² z ⁸ .

Division of monomials. A quotient of two monomials can be simplified, if a dividend and a divisor have some powers of the same letters or numerical coefficients. In this case an exponent of the power in a divisor is subtracted from an exponent of the power in a dividend; a numerical coefficient of a dividend is divided by a numerical coefficient of a divisor.
E x a m p l e :

35 a ⁴ x ³ z ⁹ : 7 a x ² z ⁶ = 5 a ³ x z ³ .

Polynomial is an algebraic sum of monomials. Degree of polynomial is the most of degrees of monomials, forming this polynomial.

Multiplication of sums and polynomials: a product of the sum of two or some expressions by any expression is equal to the sum of the products of each of the addends by this expression:

( p+ q+ r ) a = pa+ qa+ ra - opening of brackets.

Instead of the letters p, q, r, a any expressions can be taken.
E x a m p l e :

( x+ y+ z )( a+ b )= x( a+ b )+ y( a+ b ) + z( a+ b ) =
= xa + xb + ya + yb + za + zb .

A product of sums is equal to the sum of all possible products of each addend of one sum to each addend of the other sum.

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