Polynomial and Radical Functions PDF

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POLYNOMIAL AND RADICAL FUNCTIONS POLYNOMIAL FUNCTIONS (PPT 5.1) Standard Form:

terms of the polynomial function/expression by descending degrees

DEGREE

NAME

0

Constant

1

Linear

2

Quadratic

3

Cubic

4

Quartic

5

Quintic

POLYNOMIAL

NO. OF TERMS

NAME

(Monomial)

1

Monomial

(Linear Binomial)

2

Binomial

3

Trinomial

1

Monomial

4

Polynomial of 4 terms

2

Binomial

5 6y 2x² - 4y

3x³

(Quadratic Trinomial) (Cubic Monomial)

2x4+ x³ - 5x² + 11

(Quartic Polynomial)

x5 + 7x² (Quintic Binomial)

The degree of polynomial functions affects the shape of the graph and the number of turning points (points where the graph changes direction), and the end behavior (directions of the graph to the far left and far right).

End Behavior of a Polynomial Function With Leading Term axn, n Even (n ≠ 0)

n Odd

a Positive

Up and Up

Down and Up

a Negative

Down and Down

Up and Down

POLYNOMIALS, LINEAR FACTORS, ZEROS (PPT 5.2) Factor Theorem The expression x - a is a factor of a polynomial if and only if the value a is a zero of the related polynomial function.

You can write a polynomial function in factored form: g(x) = (x + 2)2 (x – 2)(x – 3). In g(x) the repeated linear factor x + 2 which makes -2 a multiple zero. Since the linear factor x + 2 appears twice, you can say that -2 is a zero of multiplicity 2.

A polynomial function can have a relative minimum (the value of the function at an up-to-down turning point) and a relative maximum (the value of the function at a down-to-up turning point).

SOLVING POLYNOMIALS EQUATIONS (PPT 5.3) To solve polynomial equations by factoring: 1. Write the equation in the form P(x) = 0 for some polynomial function P. 2. Factor P(x). Use the Zero Product Property to find the roots.

Methods in Factoring Polynomials: 1. Factoring out the GCF

2. Quadratic Trinomials

3. Perfect Square Trinomials

4. Difference of Squares

5. Factoring by Grouping

6. Sum or Difference of Cubes

Dividing Polynomials (PPT 5.4) Polynomial Long Division You can divide polynomials using steps similar to that of the long division steps used to divide whole numbers. example:

Synthetic Division A simplified version of the long-division process, (writing only the coefficients of the polynomial in standard form) for dividing linear expression x - a.

example:

The Remainder Theorem If you divide a polynomial P(x) of degree n ≥ 1 by x - a, then the remainder is P(a). example:

Theorems About Roots of Polynomial Equations (PPT 5.5) Rational Root Theorem

example of finding rational roots with a table:

example of finding synthetic division:

Conjugate Root Theorem

rational

roots

with

trial-and-error

and

example:

Descartes’ Rule of Signs P(x) = polynomial with real coefficients written in standard form -

The number of positive real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less than that by an even number - The number of negative real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(-x) or is less than that by an even number (count multiple roots according to their multiplicity.)

The Fundamental Theorem of Algebra (PPT 5.6)

-

If P(x) is a polynomial of degree n ≥>= 1, then P(x) = 0 has exactly n roots, including multiple and complex roots.

1. 2. 3. 4. 5. 6.

Find the rational root of the polynomial Make that zero an expression using the Factor Theorem Divide the polynomial function to the expression in step 2 Factorize to groups of binomials Calculate the no. of zeros No. of zeros = No. of roots

-

The degree of a polynomial equation tells you how many roots the equation has.

example of solving for the roots using graphing:

example of solving for the roots using synthetic division and

factoring:

example of solving for the roots using graphing, synthetic division, and the Quadratic Formula:

Roots and Radical Expressions (PPT 6.1)

The nth Root

example:

A. B. C.

³√0.027 = ³√0.027 = ³√(0.3)³ = 0.3 √1/25 = √1/25 = √1/(5)² = 1/5 4 √16 = 4√(4)4 = 4

nth Root of nth Powers

example: A. ³√64x³ = ³√(4x)³ = 4x B. √x4y9 = √(x²y³)² = x²y³ C.

4

√32x12y⁸ z4 = 4√(2x4y²z)4 = 2x4y²z

MULTIPLYING AND DIVIDING RADICAL EXPRESSIONS (PPT 6.2) Combining Radical Expressions: Products

-

When dividing radical expressions, ALWAYS simplify it by rationalizing the denominator example: A.

B.

BINOMIAL RADICAL EXPRESSIONS (PPT 6.3) Combining Radical Expressions: Sums and Differences Use the Distributive Property to add or subtract like radicals.

example: A. 9√2x + √2x = (9 + 1)√2x = 10√2x B. √8 + √2 - √18 = √2² × 2 + √2- √3² × 2 = 2√2 + √2 - 3√2 = (2 + 2 - 3)√2 = √2 C. (3 + 2√2)(4 + 2√2) = 12 + 6√2 + 8√2 + 8 = 20 + 14√2 D. (2 - √3)(2 + √3) = 2² + 2√3 - 2√3 - 3 = 4 - 3 = 1 E.

RATIONAL EXPONENTS (PPT 6.4)

Rational Exponent

example: A.

B.

Properties of Rational Exponents

SOLVING SQUARE ROOT AND OTHER RADICAL EQUATIONS (6.5) A.

√(5x - 1) - 2 = 0 √(5x - 1) = 2 5x - 1 = 4 5x = 5 x = 1

B.

C. What is the solution of √(2x - 3) + 4 = x? √(2x - 7) + 4 = x √(2x - 7) = x - 4 2x - 7 = (x - 4)² 2x + 7 = x² - 8x + 16 x² - 6x + 9 = 0 (x - 3)² = 0 x = 3 When should you check for extraneous solutions? When raising each side of an equation to a power....