Polynomial Functions IM 3 Ms Peralta What youll

  • Slides: 17
Download presentation
+ Polynomial Functions IM 3 Ms. Peralta

+ Polynomial Functions IM 3 Ms. Peralta

+ What you’ll learn today: n Polynomial functions. n Power of functions: examples. n

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

+ Polynomials § An expression in the form of f(x) = anxn + an-1

+ Polynomials § An expression in the form of f(x) = anxn + an-1 xn-1 + … + a 2 x 2 + a 1 x + ao where n is a non-negative integer and a 2, a 1, and a 0 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 0 Constant 5 1 Linear 3 x+2

+ Examples of Polynomials Degree Name Example 0 Constant 5 1 Linear 3 x+2 2 Quadratic X 2 – 4 3 Cubic X 3 + 3 x + 1 4 Quartic -3 x 4 + 4 5 Quintic X 5 + 5 x 4 - 7

Decide whether the function is a polynomial GUIDED PRACTICE function. If so, write it

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

Add & Subtract Polynomials Monomial: 1 term Binomial: 2 terms These are all polynomials

Add & Subtract Polynomials Monomial: 1 term Binomial: 2 terms These are all polynomials 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. 3 x + (-5 x) = [3 + (-5)]x = -2 x

Example 1: Rewrite Combine Like Terms

Example 1: Rewrite Combine Like Terms

EXAMPLE 1 Example 2: Add 3 y 3 – 2 y 2 – 7

EXAMPLE 1 Example 2: Add 3 y 3 – 2 y 2 – 7 y and – 4 y 2 + 2 y – 5 (3 y 3 – 2 y 2 – 7 y) + (– 4 y 2 + 2 y – 5) 3 y 3 – 2 y 2 – 4 y 2 – 7 y + 2 y – 5 Gather like terms 3 y 3 – 6 y 2 – 5 y – 5 Combine like terms

EXAMPLE 2 Example 3: Subtract 5 z 2 – z + 3 from 4

EXAMPLE 2 Example 3: Subtract 5 z 2 – z + 3 from 4 z 2 + 9 z – 12 (4 z 2 + 9 z – 12) – (5 z 2 – z + 3) 4 z 2 + 9 z – 12 – 5 z 2 + z – 3 Remember to distribute the – through the ( ) 4 z 2 – 5 z 2 + 9 z + z – 12 – 3 –z 2 + 10 z – 15 Gather like terms Combine like terms

GUIDED PRACTICE You try it! Find the sum 1. (t 2 – 6 t

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

+ Multiplying Polynomials There are three techniques you can use for multiplying polynomials. It’s

+ Multiplying Polynomials There are three techniques you can use for multiplying polynomials. It’s all about how you write it… 1) Distributive Property 2) FOIL 3) Box Method

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

+ 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. It is

+ 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

+ (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. y 2 + 7 y + 3 y + 21 Combine like terms. y 2 + 10 y + 21

+ Multiply (2 x - 5)(x 2 - 5 x + 4) n You

+ Multiply (2 x - 5)(x 2 - 5 x + 4) n You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2 x(x 2 - 5 x + 4) - 5(x 2 - 5 x + 4) 2 x 3 - 10 x 2 + 8 x - 5 x 2 + 25 x - 20 n Group and combine like terms. 2 x 3 - 10 x 2 - 5 x 2 + 8 x + 25 x - 20 2 x 3 - 15 x 2 + 33 x - 20

+ Multiply (2 x - 5)(x 2 - 5 x + 4) You cannot

+ Multiply (2 x - 5)(x 2 - 5 x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x 2 -5 x +4 2 x 2 x 3 -10 x 2 +8 x -5 -5 x 2 +25 x -20 Almost done! Go to the next slide!

2 - 5 x + 4) Multiply (2 x 5)(x + Combine like terms!

2 - 5 x + 4) Multiply (2 x 5)(x + Combine like terms! x 2 -5 x +4 2 x 2 x 3 -10 x 2 +8 x -5 -5 x 2 +25 x -20 2 x 3 – 15 x 2 + 33 x - 20