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