Unit+6

=Unit 6 Polynomials and Polynomial Functions=

Vocabulary

 * Binomial Theorem (p. 348) || Fundamental Theorem of Algebra (p. 335) || polynomial function (p. 301) ||
 * combination (p. 340) || Imaginary Root Theorem (p. 332) || Rational Root Theorem (p. 329) ||
 * complex conjugates (p. 332) || Irrational Root Theorem (p. 331) || relative maximum (p. 309) ||
 * conjugates (p. 331) || multiple zero (p. 310) || relative minimum (p. 309) ||
 * degree (p. 301) || multiplicity (p. 310) || Remainder Theorem (p. 317) ||
 * degree of a polynomial (p. 301) || //n// factorial (p. 339) || standard form of a polynomial (p. 301) ||
 * difference of cubes (p. 322) || Pascal's Triangle (p. 347) || sum of cubes (p. 322) ||
 * expand (p. 347) || permutation (p. 339) || synthetic division (p. 315) ||
 * Factor Theorem (p. 309) || polynomial (p. 30 ||

6-1 Objectives
A **polynomial** is a monomial or a sum of monomials with whole-number exponents. The exponent of the variable in a term is the **degree** of that term. The **degree of a polynomial** is the largest degree of any term of the polynomial. When the terms of a polynomial are in descending order by degree, the polynomial is in standard form. You can classify a polynomial by the number of terms it contains or by its degree. A **polynomial function** in one variable can be written in the form //P//(//x//) = //a////n////x////n// + //a////n// − 1//x////n// − 1 + ... + //a//1//x// + //a//0, where //n// ≥ 0 and the coefficients //a////n// , ... , //a//0 are complex numbers. You can use a calculator to find cubic or quartic polynomial functions to model data, just as you have done with linear and quadratic polynomial functions.
 * 1) To classify polynomials (p. 300)
 * 2) To model data using polynomial functions (p. 302)

6-2 and 6-3 Objectives
A polynomial can be factored into linear factors. The **Factor Theorem** states that the expression //x// − //a// is a linear factor of a polynomial if and only if //a// is a zero of the related polynomial function. Then //a// is an //x// -intercept of the polynomial function and is a solution of the related polynomial equation. If the zeros of a polynomial function are known, a polynomial function can be determined by finding the product of the corresponding linear factors. If //x// − //a// is repeated as a factor //k// times, then //a// is a **multiple zero** of the polynomial—a zero of **multiplicity** //k//. When you consider only neighboring points on a graph, the greatest //y// -value occurs at a **relative maximum** and the least //y// -value occurs at a **relative minimum.** You can divide a polynomial by one of its factors to find another factor. When you divide by a linear factor, you can simplify this division by writing only the coefficients of each term. This process is called **synthetic division.** The **Remainder Theorem** guarantees that //P//(//a//) is the remainder when //P//(//x//) is divided by //x// − //a//.
 * 1) To analyze the factored form of a polynomial (p. 307)
 * 2) To write a polynomial function from its zeros (p. 309)
 * 3) To divide polynomials using long division (p. 314)
 * 4) To divide polynomials using synthetic division (p. 315)

6-4 Objectives
You can solve polynomial equations by graphing or by factoring. The **sum of cubes** and the **difference of cubes** have factor formulas. Sometimes you can use the Quadratic Formula to factor polynomial expressions of higher degree.
 * 1) To solve polynomial equations by graphing (p. 321)
 * 2) To solve polynomial equations by factoring (p. 322)

6-5 and 6-6 Objectives
The **Rational Root Theorem** identifies all possible rational roots of a polynomial equation with integer coefficients. A rational root of a polynomial equation is the quotient of a factor of the constant term and a factor of the leading coefficient. Number pairs of the form //a// + and //a// −  are called **conjugates**, while those of the form //a// + //b////i// and //a// − //b////i// are called **complex conjugates.** The **Irrational Root Theorem** states that irrational roots of a polynomial equation with rational coefficients occur in conjugate pairs. Similarly, the **Imaginary Root Theorem** states that imaginary roots of a polynomial equation with real coefficients occur in complex conjugate pairs. The **Fundamental Theorem of Algebra** and its corollary assert that an //n// th degree polynomial equation, where //n// ≥ 1, has exactly //n// complex roots.
 * 1) To solve equations using the Rational Root Theorem (p. 329)
 * 2) To use the Irrational Root Theorem and the Imaginary Root Theorem (p. 331)
 * 3) To use the Fundamental Theorem of Algebra (p. 335)
 * 4) To solve polynomial equations with complex zeros (p. 336)

6-7 and 6-8 Objectives
The notation //n// !, read **" //n// factorial,"** means //n//(//n// − 1)(//n// − 2) • ... • 3 • 2 • 1, and 0 ! = 1 . A **permutation** is an arrangement of items in a particular order. You can count permutations using the Multiplication Counting Principle or factorial notation. To compute the number of permutations of //n// objects chosen //r// at a time, you can also use the formula //n//P//r// =, for 0 ≤ //r// ≤ //n//. A selection in which order does not matter is a **combination.** The number of combinations of //n// objects chosen //r// at a time is //n//C//r// =, for 0 ≤ //r// ≤ //n//. Use the **Binomial Theorem** to expand a binomial raised to a power. For //n// ≥ 0, (//a// + //b//)//n// = //n//C0//a////n// + //n//C1//a////n// − 1//b// +//n//C2//a////n// − 2//b//2 +... +//n//C//n// − 1//a////b////n// − 1 +//n//C//n////b////n//. The coefficients in the expansion of (//a// + //b//)//n// are found in **Pascal's Triangle.** You can also use the Binomial Theorem to find probabilities when an event has only two possible outcomes.
 * 1) To count permutations (p. 339)
 * 2) To count combinations (p. 340)
 * 3) To use Pascal's Triangle (p. 347)
 * 4) To use the Binomial Theorem (p. 348)