POLYNOMIALS

 A polynomial is a term or the sum or difference of two or more terms.  A polynomial has no variables in the denominator.  The “degree of a term” is the exponent of the variable (4x 3 is a 3 rd degree term).  The “degree of the polynomial” is the same as the degree of the term with the highest degree. (x 5 + 4x 3 – 3x + 2 is a fifth degree polynomial)  Polynomials in standard form are in order of degree from highest to lowest with the constant at the end.

POLYNOMIALS BY TERMS  A polynomial with one term is called a monomial.  A polynomial with two terms is called a binomial.  A polynomial with three terms is called a trinomial.

POLYNOMIALS BY DEGREE  A first degree polynomial is linear.  A second degree polynomial is quadratic.  A third degree polynomial is cubic.  A polynomial with no variable is called a constant.

Examples 1 and 2 Name the degree of each term and each polynomial. Put them in standard form. Degree of each term 5, 3, 1, and 0 Degree of the polynomial 5 It’s in standard form. New Problem: Degree of each term 1, 2, 0, and 3 Degree of the polynomial 3 rd In standard form, it is:

ADDING POLYNOMIALS  Collect like terms.  In order to have like terms, the variable parts must be exactly the same.  Combine the coefficients (the numbers in front of the variable).

EXAMPLES 3 AND 4  Add and put in standard form:

SUBTRACTION POLYNOMIALS  Drop the first set of parentheses.  Distribute a –1 in the second set of parentheses.  Combine like terms.

EXAMPLE 5  Subtract and put in standard form:

FINDING GCF  Put all the numbers in a single division house with vertical lines separating the numbers.  Divide by something that will go into all the numbers under the division house evenly (it does NOT have to be prime).  Continue dividing by an expression that will go into all the numbers under the division house evenly until the only thing that will divide into all of them is 1.  Multiply all the numbers on the left together.

FIND THE GCF OF 60 AND The only thing that will Go into both 3 and 5 is 1. Stop here. The GCF is 20.

FIND THE GCF OF 3X 3 AND 6X 2 3x 3 6x 2 3 x 3 2x 2 x x 2 2x x x2x2 The GCF is 3x 2

There are no common factors other than 1. The GCF of 16a 6 and 7b is 1. Find the GCF of16a 6 and 9b

FACTORING BY GCF Put the expression in a division tower Continue to divide by numbers or variables until there is no number or variable common to all terms. Put the numbers and variables along the side on the outside of the parentheses. Put the top expression on the inside of parentheses.

EXAMPLE 1  Factor: 56x 4 – 32x 3 – 72x 2

TRY THESE…  1.(3 – x 3 – 5x 2 ) + (x + 2x 3 – 3)  x 3 – 5x 2 + x  2.(x 2 + 4) – (x – 4) + (x 2 – 2x)  2x 2 - 3x + 8 Find the GCF each pair of monomials x and 28x 3  4x 4. 27x 2 and 45x 3 y 2  9x 2

TRY THESE…  Factor each polynomial. Check your answer.  5. 16x + 20x 3  4x(5x 2 + 4)  6. 4m 4 – 12m 2 + 8m  4m(2m 2 – 3m + 2) 7. A rocket is fired vertically into the air at 40 m/s. The expression –5t t + 20 gives the rocket’s height after t seconds. Factor this expression.  -5(t 2 – 8t – 4)