Polynomial Functions IM3 Ms.Peralta

What you’ll learn today: Polynomial functions. Power of functions: examples. Add and subtract polynomials. Multiply polynomials.

f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao Polynomials An expression in the form of f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao where n is a non-negative integer and a2, a1, and a0 are real numbers. The function is called a polynomial function of x with degree n. A polynomial is a monomial or a sum of terms that are monomials. Polynomials can NEVER have a negative exponent or a variable in the denominator! The term containing the highest power of x is called the leading coefficient, and the power of x contained in the leading terms is called the degree of the polynomial.

Examples of Polynomials Degree Name Example Constant 5 1 Linear 3x+2 2 Quadratic X2 – 4 3 Cubic X3 + 3x + 1 4 Quartic -3x4 + 4 Quintic X5 + 5x4 - 7

polynomial function; f (x) = –2x + 13; degree 1; type: linear; Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. GUIDED PRACTICE 1. f (x) = 13 – 2x polynomial function; f (x) = –2x + 13; degree 1; type: linear; leading coefficient: –2 2. p (x) = 9x4 – 5x –2 + 4 not a polynomial function polynomial function; h(x) = 6x2 – 3x + π ; degree 2, type: quadratic; leading coefficient: 6 3. h (x) = 6x2 + π – 3x

Add & Subtract Polynomials Monomial: 1 term These are all polynomials Binomial: 2 terms Trinomial: 3 terms Adding Polynomials: Combine the like terms Like Terms – Terms that have the same variables with the same exponents on them Combining Like Terms: Add the coefficients of each all like terms. Ex. 3x + (-5x) = [3 + (-5)]x = -2x

Example 1: Rewrite Combine Like Terms

Example 2: Add 3y3 – 2y2 – 7y and –4y2 + 2y – 5 (3y3 – 2y2 – 7y) + (–4y2 + 2y – 5) 3y3 – 2y2 – 4y2 – 7y + 2y – 5 Gather like terms 3y3 – 6y2 – 5y – 5 Combine like terms

Example 3: Subtract 5z2 – z + 3 from 4z2 + 9z – 12 (4z2 + 9z – 12) – (5z2 – z + 3) Remember to distribute the – through the ( ) 4z2 + 9z – 12 – 5z2 + z – 3 4z2 – 5z2 + 9z + z – 12 – 3 Gather like terms –z2 + 10z – 15 Combine like terms

You try it! Find the sum 1. (t2 – 6t + 2) + (5t2 – t – 8) GUIDED PRACTICE Find the sum 1. (t2 – 6t + 2) + (5t2 – t – 8) t2 + 5t2 – 6t – t + 2 – 8 6t2 – 7t – 6 Find the difference 2. (8d – 3 + 9d3) – (d3 – 13d2 – 4) 8d – 3 + 9d3 – d3 + 13d2 + 4 9d3 – d3 + 13d2 + 8d – 3 + 4 8d3 + 13d2 + 8d + 1

There are three techniques you can use for multiplying polynomials. It’s all about how you write it… Distributive Property FOIL Box Method

Remember, FOIL reminds you to multiply the: First terms Outer terms Inner terms Last terms

The FOIL method is ONLY used when you multiply 2 binomials The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. Use the FOIL method to multiply the following binomials: (y + 3)(y + 7).

(y + 3)(y + 7) F tells you to multiply the FIRST terms of each binomial. O tells you to multiply the OUTER terms of each binomial. I tells you to multiply the INNER terms of each binomial. L tells you to multiply the LAST terms of each binomial. y2 + 7y + 3y + 21 Combine like terms. y2 + 10y + 21

Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x2 - 5x + 4) - 5(x2 - 5x + 4) 2x3 - 10x2 + 8x - 5x2 + 25x - 20 Group and combine like terms. 2x3 - 10x2 - 5x2 + 8x + 25x - 20 2x3 - 15x2 + 33x - 20

Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x2 -5x +4 2x -5 2x3 -10x2 +8x Almost done! Go to the next slide! -5x2 +25x -20

Multiply (2x - 5)(x2 - 5x + 4) Combine like terms! +4 2x -5 2x3 -10x2 +8x -5x2 +25x -20 2x3 – 15x2 + 33x - 20