2-1 Operations on Polynomials

Refer to the algebraic expression above to complete the following: 1)How many terms are there? 2)Give an example of like terms. 3)Give an example of a coefficient. 4)Give an example of a constant. 5)Simplify the expression.

1)There are 6 terms – a term is a constant or a variable or a product of a constant and a variable separated by + and – signs. 2)-6, 1 and are like terms – terms with the same variable to the same power. 3)2, 7, -5, and -1 are coefficients – when the term contains a number and a variable, the number part is the coefficient. 4)-6 and 1 are constants – a term that does not have a variable.

To simplify the expression: Add the coefficients of any like terms. Keep the variable and exponent the same. Write in order with the largest exponent first which is the standard (general form) of a polynomial.

An expression formed by adding a finite number of “same base” unlike terms. Example: Exponents must be positive integers (no fractions), there can be no square roots, and no variables in the denominator. - Not a polynomial

 The exponent of a term is the degree of the term. Example has a degree of 5 The value of the largest term is the degree of a polynomial. Example has a degree of 3 The leading coefficient is the coefficient of the first term when the polynomial is written in standard form (largest degree first).

1) 2)9 + 3x 3)

Drop parentheses and add like terms. Make sure answer is in standard form

Line up terms by degree

Change the signs of the second polynomial and then add.

 If you add or subtract polynomials your answer is also a polynomial.  This means polynomials are “closed” under addition and subtraction.

2(3x – 1) Distributive property

If you multiply polynomials you get a polynomial as the answer. Polynomials are “closed” for multiplication!!!

Let’s do a very simple one. Polynomials are NOT “closed” for divison!!!

(x + 3)(2x – 1)Distributive property (twice) Combine “like” terms

Distributive property (twice) Combine “like” terms

How do you multiply 2 * 3 * 4 three numbers? 6* 4= 24 OR: 2 * 3 * *= 24OR: 2 * 3 * 4 8* 3= 24 Pick 2 factors, multiply them to get a product, then multiply the product by the last factor

Associative Property: if you have 3 or more factors, pick two, multiply them 1 st 2*3*4 (2*3)*4 (to visually show that we are picking 2, we group, or associate them together with parentheses).

(x – 1)(2x + 3)(3x – 2) = ? = [ (x – 1)(2x + 3) ] (3x – 2) How do you multiply three binomials? Pick 2 factors, multiply them to get a product, then multiply the product by the last factor  associative property.

Square of a sum. (x + y)(x + y)

Square of a difference. (x - y)(x - y)

Product of a sum and a difference. (x + y)(x – y)

Cube of a sum.

Cube of a difference.