#### Study sets matching "zeros of polynomials"

#### Study sets matching "zeros of polynomials"

factored form

Standard form

Line of symmetry

Vertex

Algebraic Expression written as the product of the factors.

Ax + By=C, where A, B, and C are not decimals or fractions, w…

a line that divides a figure into two halves that are mirror…

A point where two or more straight lines meet.

factored form

Algebraic Expression written as the product of the factors.

Standard form

Ax + By=C, where A, B, and C are not decimals or fractions, w…

Dividend

Divisor

Quotient

Long Division

A number to be divided by another number

A number by which another number is to be divided

The result of a division problem

Works every time

Dividend

A number to be divided by another number

Divisor

A number by which another number is to be divided

f(x) = (x+1)²(x-3)⁵

f(x) = (x-1)²(x+2)³

f(x) = (x+1)³(x+6)⁴

f(x) = (x-3)²

Bounces off @ -1... Crosses @ 3

Bounces off @ 1... Crosses @ -2

Crosses @ -1... Bounces off @ -6

Bounces off @ 3

f(x) = (x+1)²(x-3)⁵

Bounces off @ -1... Crosses @ 3

f(x) = (x-1)²(x+2)³

Bounces off @ 1... Crosses @ -2

Explain the "Affenkasten".

What can multiple zeros of a polynomia…

What is a zeros's "multiplicity"?

What if many graphs have the same zeros?

All cubed polynomials have a point of inflection (Wendepunkt)…

You can find how many times the graph is going to touch or cr…

It refers to the number of times its associated factor appear…

They can still look very different depending on what exponent…

Explain the "Affenkasten".

All cubed polynomials have a point of inflection (Wendepunkt)…

What can multiple zeros of a polynomia…

You can find how many times the graph is going to touch or cr…

1/2, 5

−5/2, 3, 4

1/2, 1, −1

0, 1, 1

Find the zeros of f(x) = (2x − 1)(x − 5)

Find the zeros of f(x) = (2x + 5)(x² − 7x + 12)

Find the zeros of f(x) = 2x³ − x² − 2x + 1

Find the zeros of f(x) = x³ − 2x² + x

1/2, 5

Find the zeros of f(x) = (2x − 1)(x − 5)

−5/2, 3, 4

Find the zeros of f(x) = (2x + 5)(x² − 7x + 12)

1/2, 5

−5/2, 3, 4

0, 1/2, 1, −1

0, 1 mult. 2

Find the zeros of f(x) = (2x − 1)(x − 5)

Find the zeros of f(x) = (2x + 5)(x^2 − 7x + 12)

Find the zeros of f(x) = x(2x − 1)(x − 1)(x + 1)

Find the zeros of f(x) = x^3 − 2x^2 + x

1/2, 5

Find the zeros of f(x) = (2x − 1)(x − 5)

−5/2, 3, 4

Find the zeros of f(x) = (2x + 5)(x^2 − 7x + 12)

Long Division

Synthetic Division

Remainder Theorem

Factor Theorem

This form of division requires the division of a larger funct…

This form of division also requires the division of a larger…

If a polynomial is divided by x - k, the remainder is r = f(k)

A polynomial has a factor (x - k) if and only if (iff) f(k) = 0

Long Division

This form of division requires the division of a larger funct…

Synthetic Division

This form of division also requires the division of a larger…

3

-3

16

2

If (x-3) is a factor, then x = ____ is a zero.

If P(-3) = 0, then x = ____ is a root.

if P(x) = 2(x-2)(x-4) then the y-int is ____ .

if P(x) = 2(x-2)(x-4) then the degree is ____ .

3

If (x-3) is a factor, then x = ____ is a zero.

-3

If P(-3) = 0, then x = ____ is a root.

.5, 5

−2.5, 3, 4

0, .5, 1, −1

0, 1

Find the zeros of f(x) = 2x² - 11x + 5

Find the zeros of f(x) = 2x³ - 9x² - 11x + 60

Find the zeros of f(x) = 2x⁴ - x³ - 2x² + x

Find the zeros of f(x) = x³ - 2x² +x

.5, 5

Find the zeros of f(x) = 2x² - 11x + 5

−2.5, 3, 4

Find the zeros of f(x) = 2x³ - 9x² - 11x + 60

finding real zeros

solving inequalities

finding complex 0's

solving over complex

plug in calc, if there are not enough, then do synthetic divi…

set to 0, graph it, remember open and closed dots q

plug in calc, use zeros to synetic divide, write out factored…

synthetic divide, then complete the square/factor/quadratic f…

finding real zeros

plug in calc, if there are not enough, then do synthetic divi…

solving inequalities

set to 0, graph it, remember open and closed dots q

Division Algorithm fro Polynomials

Fraction Form

d(x)

Possible Rational Zeros

f(x)=d(x)*q(x)+r(x)

f(x)/d(x)=q(x)=r(x)/d(x)

divisor

p/q, where p is the constant and q is the coefficient of the…

Division Algorithm fro Polynomials

f(x)=d(x)*q(x)+r(x)

Fraction Form

f(x)/d(x)=q(x)=r(x)/d(x)

X = 1 ± i ; P(x) = (x - 1 - i) (x - 1…

X = ± √5, ± i √5 ;... P(x) = (x - √5) (x…

1 + 3i, 2 - 5i

i, 1 + i

Find all zeros (real and complex). Factor polynomial as a pro…

Find all zeros (real and complex). Factor polynomial as a pro…

Find all remaining zeros... 13) Degree: 6 Zeros: 2(multiplicity…

Find all remaining zeros... 15) Degree: 6 Zeros: -i, 1-i (multi…

X = 1 ± i ; P(x) = (x - 1 - i) (x - 1…

Find all zeros (real and complex). Factor polynomial as a pro…

X = ± √5, ± i √5 ;... P(x) = (x - √5) (x…

Find all zeros (real and complex). Factor polynomial as a pro…

Find all zeros (real and complex). Fac…

Find all zeros (real and complex). Fac…

Find all remaining zeros... 13) Degree:…

Find all remaining zeros... 15) Degree:…

X = 1 ± i ; P(x) = (x - 1 - i) (x - 1 + i)

X = ± √5, ± i √5 ;... P(x) = (x - √5) (x + √5) (x - i √5) (x +…

1 + 3i, 2 - 5i

i, 1 + i

Find all zeros (real and complex). Fac…

X = 1 ± i ; P(x) = (x - 1 - i) (x - 1 + i)

Find all zeros (real and complex). Fac…

X = ± √5, ± i √5 ;... P(x) = (x - √5) (x + √5) (x - i √5) (x +…

Quadratic equations

Example: Find the zeroes for the funct…

1. Write the polynomial in factored fo…

2. What are the zeroes of the function…

Quadratic equations are a type of polynomial, so it makes sen…

1. From the factored form, the zeroes can be found by setting…

B.) 4x(x-4)(x+6

D.) 0, -2, -5

Quadratic equations

Quadratic equations are a type of polynomial, so it makes sen…

Example: Find the zeroes for the funct…

1. From the factored form, the zeroes can be found by setting…

monomial

degree (of a monomial)

binomial

trinomial

a product of a number and a finite number of variables

the sum of the powers of the variables

sum of two monomials

sum of three monomials

monomial

a product of a number and a finite number of variables

degree (of a monomial)

the sum of the powers of the variables

binomial

coefficient

degree of a polynomial

polynomial

a polynomial with two terms

A number multiplied by a variable in an algebraic expression

The greatest degree of any term in the polynomial

A monomial or a sum or difference of monomials

binomial

a polynomial with two terms

coefficient

A number multiplied by a variable in an algebraic expression

polynomial

polynomial in one variable

degree of a polynomial

leading coefficient

monomial or a sum of monomials

a polynomial that only contains one variable

greatest exponent of its variable

coefficient of the term with the highest degree

polynomial

monomial or a sum of monomials

polynomial in one variable

a polynomial that only contains one variable

Polynomial

Standard Form

Monomial

Binomial

A mathematical expression consisting of a sum of terms, each…

Order terms from highest to lowest exponents

Polynomial with exactly one term

Polynomial with exactly two terms

Polynomial

A mathematical expression consisting of a sum of terms, each…

Standard Form

Order terms from highest to lowest exponents

Find the zeros and their multiplicitie…

Find the zeros and their multiplicitie…

Find the zeros and their multiplicitie…

Find the zeros and their multiplicitie…

-2(mult of 3)

0(mult of 1)... 3(mult of 4)

4(mult of 2)

2(mult of 2)... -5(mult of 3)

Find the zeros and their multiplicitie…

-2(mult of 3)

Find the zeros and their multiplicitie…

0(mult of 1)... 3(mult of 4)

Polynomial

Quartic Function

Quintic Function

Rational expression

A monomial or a sum or difference of monomials

a polynomial function of degree four

a polynomial function of degree five

a fraction with algebraic expressions in the numerator and de…

Polynomial

A monomial or a sum or difference of monomials

Quartic Function

a polynomial function of degree four

Monomial

Degree of a monomial

Polynomial

Degree of a polynomial

A number, a variable, or a product of numbers and variables w…

The sum of the exponents of the variables.

A monomial or a sum or difference of monomials.

The degree of the term with the greatest degree.

Monomial

A number, a variable, or a product of numbers and variables w…

Degree of a monomial

The sum of the exponents of the variables.

monomial

degree (of a monomial)

binomial

trinomial

a product of a number and a finite number of variables

the sum of the powers of the variables

sum of two monomials

sum of three monomials

monomial

a product of a number and a finite number of variables

degree (of a monomial)

the sum of the powers of the variables

Coefficient

degree

end behavior

Fundamental Theorem of Algebra

a number multiplied by a variable

the greatest exponent of its variable

the value of f(x) as x approaches positive and negative infin…

every non-zero single variable polynomial with complex coeffi…

Coefficient

a number multiplied by a variable

degree

the greatest exponent of its variable

x = -3 and x = 2

x = -2 and x = 3

x = -2 and x = -3

x = 2 and x = 3

Determine the solution of the polynomial graph.

Determine the solution of the polynomial graph.

Determine the solution of the polynomial graph.

Determine the solution of the polynomial graph.

x = -3 and x = 2

Determine the solution of the polynomial graph.

x = -2 and x = 3

Determine the solution of the polynomial graph.

x = -3 and x = 2

x = -2 and x = 3

x = -2 and x = -3

x = 2 and x = 3

Determine the solution of the polynomial graph.

Determine the solution of the polynomial graph.

Determine the solution of the polynomial graph.

Determine the solution of the polynomial graph.

x = -3 and x = 2

Determine the solution of the polynomial graph.

x = -2 and x = 3

Determine the solution of the polynomial graph.

Standard form of a polynomial

Term of a polynomial

monomial

binomial

a polynomial that is written with the terms in order from gre…

A collection of numbers and/or letters multiplied together. T…

an expression with only one term

an expression with two terms

Standard form of a polynomial

a polynomial that is written with the terms in order from gre…

Term of a polynomial

A collection of numbers and/or letters multiplied together. T…

Polynomial

Degree

Leading Coefficient

End Behavior

an expression consisting of numbers, variables and non-negati…

Found by identifying the largest exponent.

The coefficient of the term with the largest exponent

What the y-values do as x approaches positive and negative in…

Polynomial

an expression consisting of numbers, variables and non-negati…

Degree

Found by identifying the largest exponent.

Term

Degree of a monomial

Degree of a polynomial

Binomial

A number, a variable, or a product or quotient of a number an…

The sum of the exponents on all the variables

The degree of the term with the highest degree in the polynom…

A polynomial consisting of two terms

Term

A number, a variable, or a product or quotient of a number an…

Degree of a monomial

The sum of the exponents on all the variables

Standard Form

Terms

Binomial

Trinomial

terms in order from the greatest degree to the least degree

a mathematical expression

a polynomial with two terms

a polynomial with three terms

Standard Form

terms in order from the greatest degree to the least degree

Terms

a mathematical expression

Polynomial Functions ... (2 restrictions…

Polynomial Functions... (3 restrictions o…

Parts of a Polynomial... (8 parts)

Answers to a Polynomial Equations... (3 n…

1. Lead coefficient cannot be zero... 2. All coefficients must b…

1. Non-negative... 2. Integers... 3. Listed greatest to least

1. Lead Coefficient... 2. Degree... 3. Term... 4. Cubic Term... 5. Quadra…

1. Roots... 2. Solutions... 3. Zeros

Polynomial Functions ... (2 restrictions…

1. Lead coefficient cannot be zero... 2. All coefficients must b…

Polynomial Functions... (3 restrictions o…

1. Non-negative... 2. Integers... 3. Listed greatest to least

Polynomials

Degree

1st degree

2nd degree

Expressions that have no operations other than addition, subt…

The degree of a polynomial with one variable is the exponent…

linear polynomial (17x+4)

Quadratic polynomial (9x^2+2x+11)

Polynomials

Expressions that have no operations other than addition, subt…

Degree

The degree of a polynomial with one variable is the exponent…