Introduction to Polynomials.ppt0..............................
- 2. Monomial: A number, a variable or the product of a number and one or more variables. Polynomial: A monomial or a sum of monomials. Binomial: A polynomial with exactly two terms. Trinomial: A polynomial with exactly three terms. Coefficient: A numerical factor in a term of an algebraic expression.
- 3. Degree of a monomial: The sum of the exponents of all of the variables in the monomial. Degree of a polynomial in one variable: The largest exponent of that variable. Standard form: When the terms of a polynomial are arranged from the largest exponent to the smallest exponent in decreasing order.
- 4. What is the degree of the monomial? 2 4 5 b x The degree of a monomial is the sum of the exponents of the variables in the monomial. The exponents of each variable are 4 and 2. 4+2 = 6. The degree of the monomial is 6. The monomial can be referred to as a sixth degree monomial.
- 5. A polynomial is a monomial or the sum of monomials 2 4x 8 3 3 x 14 2 5 2 x x Each monomial in a polynomial is a term of the polynomial. The number factor of a term is called the coefficient. The coefficient of the first term in a polynomial is the lead coefficient. A polynomial with two terms is called a binomial.
- 6. 1 4 x 8 3 3 x 14 2 5 2 x x The degree of a polynomial in one variable is the largest exponent of that variable. 2 A constant has no variable. It is a 0 degree polynomial. This is a 1st degree polynomial. 1st degree polynomials are linear. This is a 2nd degree polynomial. 2nd degree polynomials are quadratic. This is a 3rd degree polynomial. 3rd degree polynomials are cubic.
- 7. Classify the polynomials by degree and number of terms. Polynomial a. b. c. d. 5 4 2 x x x 2 3 1 4 2 3 x x Degree Classify by degree Classify by number of terms Zero Constant Monomial First Linear Binomial Second Quadratic Binomial Third Cubic Trinomial
- 8. To rewrite a polynomial in standard form, rearrange the terms of the polynomial starting with the largest degree term and ending with the lowest degree term. The leading coefficient, the coefficient of the first term in a polynomial written in standard form, should be positive.
- 9. 7 4 5 2 4 x x x x 5 4 4x 2 x 7 Write the polynomials in standard form. 2 4 3 5 5 7 2 x x x x 3 2x 4 x 7 x 5 2 5x ) 7 5 5 2 ( 1 2 3 4 x x x x 3 2x 4 x 7 x 5 2 5x Remember: The lead coefficient should be positive in standard form. To do this, multiply the polynomial by –1 using the distributive property.
- 10. Write the polynomials in standard form and identify the polynomial by degree and number of terms. 2 3 2 3 7 x x 1. 2. x x 2 3 1 2
- 11. 2 3 2 3 7 x x 2 3 2 3 7 x x 3 3x 2 2x 7 7 2 3 1 2 3 x x 7 2 3 2 3 x x This is a 3rd degree, or cubic, trinomial.
- 12. x x 2 3 1 2 x x 2 3 1 2 2 3x x 2 1 This is a 2nd degree, or quadratic, trinomial.