ch 5 polynomial functions

Factor Theorem

the expression x − a is a factor of a polynomial only if "a" is a zero of the polynomial function

degree of polynomial (in one variable)

the greatest degree of its monomial terms

x intercept of a polynomial function

the point(s) where the graph crosses the x axis. This is also called the root(s) or zero(s)

end behavior for a polynomial function: a positive and n is even

up and down

end behavior for a polynomial function: a positive and n is even

up and up

The remainder theorem

used to evaluate a polynomial with degree ≥ 1 -i.e. to find P(3), take the polynomial and divide by x − 3 -the remainder = P(3)

The Binomial Theorem

used to expand a binomial raised to nth power (a + b)ⁿ -nth row of Pascal's triangle shows coefficients -powers of a start at n and go down -powers of b start at 0 and go up

relative maximum and relative minimum

when graph of a polynomial function has several turning points: relative max is value of function at up to down turning point relative min is value of function at down to up turning point

quadratic formula

−b ± √ (b² − 4ac) ----------------------- 2a

degree of a monomial (in one variable)

exponent of the variable

methods to solve quadratic functions

- by factoring, zero prod property, only real solutions - use quadratic formula, real and non-real solutions - by graphing, find zeros, x intercepts - complete the square

Steps to factoring

1. factor out GCF, if there is one 2. if 4 terms, consider factor by grouping 3. check of special cases - difference of 2 squares - difference or sum of 2 cubes 4. quadratic trinomial, find factors x game

steps to long division for polynomials

1. write divisor & dividend in standard form, fill in zeros for missing terms 2. Divide 1st term of divisor by first term of dividend to get first term of the quotient, place above like term 3. Take the term found in step 1 and multiply it times the divisor 4. distribute neg 1 then add this with the line above 5. Repeat until done, if there is a remainder place it over divisor after the solution

rational root theorem

If P(x) is a polynomial function with integer coefficients, the only possible rational roots: factor or constant term ------------------------------------ factor of leading coefficient *creates a finite list of possible rational roots of a polynomial equation with rational coefficients

polynomial

a monomial or a sum or difference of monomials.

polynomial function of x

a polynomial with the variable x

first step in factoring

factor out the GCF, if there is one

The (n+1) Point Principle

-any 2 pts determine a unique line -any 3 pts (not on a line) determine a unique parabola -any 4 pts (not on a line or parabola) determine a unique cubic and so forth

Names of polynomials using degree

0 - constant, 1 - linear, 2 - quadratic, 3 - cubic, 4 - quartic, 5 - quintic

Names of polynomials using number of terms

1 - monomial or polynomial or 1 term 2 - binomial or polynomial of 2 terms 3 - trinomial or polynomial of 3 terms 4 - polynomial of 4 terms

steps synthetic division

1 : set the denominator equal to zero to find the number to put in the division box. 2. dividend in descending order listing only coefficients, use 0 for missing terms 3. bring the leading coefficient (first number) straight down. 4. multiply the number in the division box with the number you brought down and put the result in the next column. 5. Add the two numbers together and write the result in the bottom of the row. 6. Repeat steps 3 and 4 until you reach the end of the problem. 7. Write the final answer. The final answer is made up of the numbers in the bottom row with the last number being the remainder and the remainder must be written as a fraction. The variables or x's start off one power less than the original denominator and go down one with each term.

how to find roots of 3rd degree or higher polynomial (with a graphing calculator)

1. fundamental theorem of alg will tell you # of roots 2. use graph calc to id any real roots 3. factor out the factors using synthetic division 4. solve resulting quadratic using easiest method

finding rational roots of a polynomial

1. identify the possible roots , using rational root theorem 2. find 1 rational root substitute possible roots in, see which causes P(x) = 0 or use remainder theorem, synthetic division to see which has a remainder of zero 3. factor, divide by the root using synthetic division until you get a quadratic. 4. factor or use quadratic formula to find roots of quadratic

given that the x's are all consecutive, how can you find the degree of the polynomial function by using the differences of the y's

1st difference is constant - degree is 1, linear 2nd difference is constant - degree is 2, quadratic 3rd difference is constant - degree is 3, cubic 4th difference is constant - degree is 4, quartic and so on

conjugate root theorem

If a polynomial P(x) with rational coefficients, then irrational roots that have the form a + √b occur in conjugate pairs. a − √b is also a root If a polynomial P(x) with real coefficients, then complex roots occur in conjugate pairs. if a + bi is a complex root , a − bi is also a root

monomial

a real number, a variable, or the product of these

multiplicity

a root that appears k times has a multiplicity of k ie. p(x) = (x − 2)²(x + 4) 2 is a zero with multiplicity of 2 −4 is a zero with multiplicity of 1

Pascal's Triangle

a triangular array of numbers in which the first and the last number of each row is 1. each of other numbers is sum of the 2 numbers above it row #0.........1 row #1.........1 1 row #2.........1 2 1 row #3.........1 3 3 1 row #4.........1 4 6 4 1 row #5..........1 5 10 10 5 1

standard form of polynomial function

arranges the terms by degree in descending numerical order ex. P(x)=4x³(cubic term)+3x²(quadratic term)+5x(linear term)-2(constant term)

number of turning points for a polynomial function with a degree n (n ≥ 1) is generally....

at most n − 1 turning points

factoring by grouping

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b) ( x + y) x³ + 2x − 3x − 6 = x²(x + 2) + (−3)(x + 2) = (x² − 3) ( x + 2)

factoring perfect square trinomials

a² + 2ab + b² = (a + b)² x² + 10x + 25 = (x + 5)² or a² − 2ab + b² = (a − b)² x² − 12x + 36 = (x − 6)²

add of two squares a² + b²

a² + b² (√a + √b i)(√a − √b i)

factoring difference of 2 squares

a² − b² = (a + b)(a − b) 4x² − 25 = (2x + 5)(2x − 5)

factoring sum of 2 cubes

a³ + b³ = (a + b) (a² − ab + b²) 8x³ + 1 = (2x + 1)(4x² − 2x + 1)

factoring difference of 2 cubes

a³ − b³ = (a − b) (a² + ab + b²) 8x³ + 1 = (2x − 1)(4x² + 2x + 1)

end behavior of a polynomial function

direction the graph goes away from the origin, starting with the left side and then the right side

end behavior for a polynomial function: a negative and n is even

down and down

end behavior for a polynomial function: a positive and n is odd

down and up

The fundamental Theorem of Algebra

every polynomial equation of degree n ≥ 1 -has exactly n roots including multiple and complex roots -has n linear roots -has at least one complex zeros

zero product property

if the product of 2 (or more) factors is zero, then one or both (or more than one of the) factors must equal zero.

how does multiplicity affect the graph

multiplicity of 1 will be close to linear multiplicity of 2 will be close to quadratic multiplicity of 3 will be close to cubic and so on

turning points of polynomial function

places where the graph changes direction. the degree determines the maximum number of these.

how to find roots of 3rd degree or higher polynomial (without a graphing calculator)

1. fundamental theorem of alg will tell you # of roots 2. identify possible roots (rational root theorem) 3. identify true root(s) using synthetic division 4. solve resulting quadratic using easiest method

Root

A root of an equation is a value that, when substituted for the unknown quantity, satisfies the equation. A root is a solution of an equation. It is an x-intercept of the related function, which is why it can be called a zero. If (x 2 a) is a factor of a polynomial, then a is a root of that polynomial.

synthetic division

A shorthand method of dividing by a linear binomial of the form (x-a) by writing only the coefficients of the polynomials.

Descartes Rule of Signs

Let P(x) be a 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.)

how to find a particular row n in Pascal's triangle using graphing calculator

Stat, put in L1 0 through n L2, go up , n nCr L1 nCr is found under math, Prb