Polynomial Operations

Polynomial comes from poly- (meaning "many") and - nomial (in this case meaning "term")... so it says "many terms" example of a polynomial, this one has 3 terms

Parts A polynomial can have: constants (like 3, -20, or ½) variables (like x and y) exponents (like the 2 in y 2 ), but only positive integers are allowed One-term polynomial = MONOMIAL Two-term polynomial = BINOMIAL Three-term polynomial = TRINOMIAL

Appearance Notice: the TERMS are always separated by “+” or “ – ” signs.

These are polynomials:  3x  x – 2  -6y 2 – (7/9)x  3xyz + 3xy 2 z – 0.1xz – 200y  512v w 5  5 Standard form of a polynomials o c n x n + c n – 1 x n – 1 + …+ c 1 x + c 0

IMPORTANT Concepts The degree of a term is the sum of the exponents on the variables contained in the term. The degree of the polynomial is the largest degree of all its terms. Note that the standard form of a polynomial that is shown above is written in descending order. This means that the term that has the highest degree is written first, the term with the next highest degree is written next, and so forth.

Find the degree of the term -6x 5. Find the degree of the term 8. Find the degree of the term 4ab 3. Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: 4x 2 – 7x + 2 7x 3 – 9x 6

Find the degree of the term -6x 5. 5 Find the degree of the term 8. 0 Find the degree of the term 4ab 3. 4 (add the exponents of the variables) Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these: 4x 2 – 7x + 2 2: trinomial 7x 3 – 9x 6 6: binomial

IMPORTANT Concepts Combining polynomials involves:  ADDING or SUBTRACTING Combining requires LIKE TERMS… Recall that like terms are terms that have the exact same variables raised to the exact same exponents.

Perform the indicated operation and simplify. 5x 2 + 7x – 2x 2 – 10x + 5 (7x 5 + 4x 3 – 2x) + (-8x 5 + 4x) (5x 2 – 4x + 10) + (3x 2 – 2x – 12) (5x – 2y + 1) – (2x – 7y + 4) (8y 5 – y – 2) – (7y 5 + 5y – 4)

Perform the indicated operation and simplify. 5x 2 + 7x – 2x 2 – 10x + 5 3x 2 – 3x + 5 (7x 5 + 4x 3 – 2x) + (-8x 5 + 4x) -x 5 + 4x 3 + 2x (5x 2 – 4x + 10) + (3x 2 – 2x – 12) 8x 2 – 6x – 2 (5x – 2y + 1) – (2x – 7y + 4) 3x + 5y – 3 (8y 5 – y – 2) – (7y 5 + 5y – 4) x 5 – 6y + 2

IMPORTANT Concepts Multiplying polynomials involves:  Monomial × monomial  Monomial × binomial  Monomial × polynomial  Binomial × binomial  Binomial × polynomial

Monomial × monomial  In this case, there is only one term in each polynomial. You simply multiply the two terms together. Monomial × binomial  In this case, there is only one term in one polynomial and two terms in the other. You need to distribute the monomial to EVERY term of the binomial.

Monomial × Polynomial  In this case, there is only one term in one polynomial and more than one term in the other. You need to distribute the monomial to EVERY term of the other polynomial. Binomial × binomial  In this case, both polynomials have two terms. You need to distribute both terms of one polynomial times both terms of the other polynomial (the FOIL Method).

Binomial × Polynomial  As mentioned above, use the distributive property until every term of one polynomial is multiplied by every term of the other polynomial. Make sure that you simplify your answer by combining any like terms.

Multiplying polynomials involves:  Monomial × monomial (5x 2 y)(-7xyz 2 ) = -35x 3 y 2 z 2  Monomial × binomial (5xy)(x + y 2 ) = 5x 2 y +5xy 3  Monomial × polynomial (4x 3 )(x 2 – 2x – 12) = 4x 5 – 8x 4 – 48x 3

Multiplying polynomials involves:  Binomial × binomial (3xy – 5)(5xy + 3) = Use FOIL Method First = (3xy)(5xy) = 15x 2 y 2 Outer = (3xy)(3) = 9xy Inner = (-5)(5xy) = -25xy Last = (-5)(3) = -15 Combine like terms: 15x 2 y 2 + 9xy – 25xy – 15 Simplified answer = 15x 2 y 2 – 16xy – 15

Multiplying polynomials involves:  Binomial × polynomial (x 2 – 3)(x 2 + 7x – 2y + 9) Use the distributive property until every term of one polynomial is multiplied by every term of the other polynomial. (x 2 – 3)(x 2 + 7x – 2y + 9) (x 2 )(x 2 + 7x – 2y + 9) = x 4 + 7x 3 – 2x 2 y + 9x 2 (-3)(x 2 + 7x – 2y + 9) = -3x 2 – 21x + 6y – 27 Combine like terms: x 4 + 7x 3 – 2x 2 y + 9x 2 + (-3x 2 – 21x + 6y – 27) = x 4 + 7x 3 – 2x 2 y + 6x 2 – 21x + 6y – 27

Special Binomials Perfect Trinomial Squares o (a – b) 2 = a 2 – 2ab + b 2 o (a + b) 2 = a 2 + 2ab + b 2 Difference of Two Squares o (a – b)(a + b) = a 2 – b 2

EXAMPLES  Multiply: (x – 2y)(n + 3m)  Multiply: (x – 4)(x + 3)  Multiply: (2y – 5)(y + 3)

EXAMPLES  Multiply: (6xy – m)(6xy + m)  Multiply: (3a – 5) 2  Multiply: (2a + 1) 2  Multiply: (2ax – 3by)(2ax + 3by)