Properties of Polynomials. Polynomials are sums of "variables and exponents" expressions. Each piece of the polynomial that is being added, is called.

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Properties of Polynomials

Polynomials are sums of "variables and exponents" expressions. Each piece of the polynomial that is being added, is called a "term". – Monomials3, 4x, 3x 2, 7x 2 y, 18xyz the terms can be single or they can be a product of more than one there are no addition or subtraction signs – Binomials 3x+6, 4xy + 7z, 7x 2 y - 18xyz Two terms are separated by an addition or subtraction sign – Polynomials 4xy + 7z - 7x 2 y - 18xyz More than two terms are separated by addition and/or subtraction signs

Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); – there are no square roots of variables, – there are no fractional powers – there are no variables in the denominator of any fractions.

6x –2 This is NOT a polynomial term the variable has a negative exponent. This is NOT a polynomial term the variable is inside a radical. 4x 2 This IS a polynomial term it obeys all the rules. This is NOT a polynomial term the variable is in the denominator. 1 x 2

The exponent on a term tells you the "degree" of the term. The leading term in the above polynomial is a "second-degree term" or "a term of degree two". The second term is a "first degree" term. The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a "second-degree polynomial". This polynomial has four terms, including a fifth-degree term, a third-degree term, a first- degree term, and a constant term. The degree of the leading term tells you the degree of the whole polynomial. This is a fifth-degree polynomial. Give the degree of the following polynomial: 2x 5 – 5x 3 – 10x + 9

Simplify (5x 2 )(–2x 3 ) Simplify –3x(4x 2 – x + 10) Simplify (x + 3)(x + 2) Multiply coefficients Multiply variables (add exponents) Distribute the monomial through the trinomial Multiply coefficients and then multiply variables (add exponents) WATCH YOUR SIGNS FOIL OR multiply one above the other

Simplify (4x 2 – 4x – 7)(x + 3) Simplify (x + 2)(x 3 + 3x 2 + 4x – 17)

Simplify (3x 2 – 9x + 5)(2x 2 + 4x – 7) Simplify (x 3 + 2x 2 + 4)(2x 3 + x + 1)