What is a monomial In one variable; is the product of a constant and a variable raised to a non negative integer power. The form is axk a is the constant x is the variable k is the degree but must be ≥ 0 and an integer.

Polynomial Functions Polynomial is a monomial or a sum of monomials Some terms to know: Polynomial is a monomial or a sum of monomials Polynomial Function: function in the form f(x) = anxn + an-1xn-1 + … a0. an is the leading coefficient n is the degree, and a0 is the constant. What is the degree, leading coefficient, and constant: 5x3 + 4x – 3 Degree is 3, leading coefficient is 5 and constant is -3

Simplifying Polynomials Adding and subtracting polynomials: Only add or subtract the coefficients of terms with the same base and same exponent. (3y2 -6y4 + 5 – 6y) + (5y4 – 6y2 + 4y) -y4 – 3y2 – 2y + 5 (8y4 -2y2 + y – 4) – (3y3 – 12y2 + 8y) -3y3 + 12y2 – 8y 8y4 – 3y3 +10y2 -7y – 4

Practice adding polynomials: (x3 + 3x2 + 2) + (x2 – 4x + 4) Practice subtracting polynomials: (x2 – 3x – 4) – (x3 – 3x2 + x + 5)

Multiplying Polynomials Distribute each term to each term in the other parenthesis then combine like terms. (x + 4)3 Cube the first term; square the first term and multiply by the second term and 3; Square the second term multiply by the first term and 3; Cube the second term. (5c2 – 4)(2c2 + c – 3) 10c4 + 5c3 – 15c2 -8c2 -4c +12 x3 + 3((x)2 4) + 3(x(4)2) + 43 x3 + 12x2 + 48x + 64 10c4 + 5c3 – 23c2 – 4c + 12

More examples (x – 3)(x – 3) x2 – 6x + 9 5c2 + 7c + 1 -2c2 + 6c – 8 * You can’t just square the x and the 3 you will miss the middle term *remember distribute the negative to each term.

Practice Use multiplication property to simplify the polynomial: (3x + 1) (2x + 1) Use the rule for exponents to simplify the polynomial: (x + 1)3

Practice Use multiplication and subtraction property to simplify the polynomial: 8(4x3 – 3x2 – 1) – 6(4x3 + 8x – 2)