Study sets matching "zeros of polynomials"

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Study sets matching "zeros of polynomials"

11 terms
Zeroes and 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…
14 terms
Polynomials: Degree, Zeros, & Multiplicity
What is the degree of y = -2x+5 ?
What is the degree of f(x)=x^4 + 2x^3 ?
What is the degree of y= (x+1)(x-4)(x+…
What is the degree of y = [ (x+5)^2][(…
1
4
3
11
What is the degree of y = -2x+5 ?
1
What is the degree of f(x)=x^4 + 2x^3 ?
4
14 terms
Polynomials: Degree, Zeros, & Multiplicity
What is the lead coefficient of... y = 3…
What is the degree of ... f(x)=x⁴ + 2x³ ?
What is the degree of... y= (x+1)(x-4)(x…
What is the degree of ... y = (x+5)²(x-8)…
-7
4
3
11
What is the lead coefficient of... y = 3…
-7
What is the degree of ... f(x)=x⁴ + 2x³ ?
4
12 terms
Zeros of Polynomial Functions
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
15 terms
Zero's of Polynomials
f(x) = (x+1)²(x-3)⁵
f(x) = (x+2)³(x-4)
f(x) = (x-1)²(x+2)³
f(x) = (x+1)³(x+6)⁴
Touches @ -1... Crosses @ 3
Crosses @ -2, 4
Touches @ 1... Crosses @ -2
Crosses @ -1... Touches @ -6
f(x) = (x+1)²(x-3)⁵
Touches @ -1... Crosses @ 3
f(x) = (x+2)³(x-4)
Crosses @ -2, 4
Zero's of Polynomials
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
Zero's of Polynomials
f(x) = (x+1)²(x-3)⁵
f(x) = (x+2)³(x-4)
f(x) = (x-1)²(x+2)³
f(x) = (x+1)³(x+6)⁴
Touches @ -1... Crosses @ 3
Crosses @ -2, 4
Touches @ 1... Crosses @ -2
Crosses @ -1... Touches @ -6
f(x) = (x+1)²(x-3)⁵
Touches @ -1... Crosses @ 3
f(x) = (x+2)³(x-4)
Crosses @ -2, 4
5 terms
Match the Polynomial to Its Zeros
f(x) = 2(x-3)(x+4)
f(x) = 3x(x-2)(x+4)
f(x) = x(x-2)²
f(x) = (x-2)(x+2)
3, -4
0, 2, -4
0, 2
2, -2
f(x) = 2(x-3)(x+4)
3, -4
f(x) = 3x(x-2)(x+4)
0, 2, -4
10 terms
Math 4 Polynomials & Multiple Zeros
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…
Finding the Zeros of a Polynomial Function Algebraically
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)
12 terms
Finding the Zeros of a Polynomial Function Algebraically
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)
5 terms
2.3 - Real Zeros of Polynomial Functions
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…
12 terms
Polynomials x-ints y-ints zeros and factors
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.
Finding the Zeros of a Polynomial Function Algebraically
.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
pre calc polynomials and zeros test help
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
2.4 Real Zeroes of Polynomial Functions
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)
Alg 34 Ch 5 Polynomials: Degree, Zeros, & Multiplicity
What is the degree of y = -2x+5 ?
What is the degree of f(x)=x^4 + 2x^3 ?
What is the degree of y= (x+1)(x-4)(x+…
What is the degree of y = [ (x+5)^2][(…
1
4
3
11
What is the degree of y = -2x+5 ?
1
What is the degree of f(x)=x^4 + 2x^3 ?
4
13 terms
Math 2.5 Polynomial Functions; Complex and Real zeros
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…
13 terms
Math 2.5 Polynomial Functions; Complex and Real zeros
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 +…
7 terms
Algebra 2B : Lesson 2, Unit 1 /// Polynomials, Linear Factors, and Zeroes
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…
Polynomials
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
Chapter 9: Polynomials
Monomial
Binomial
Trinomial
Degree of a Monomial
a polynomial with one term
a polynomial with two terms
a polynomial with three terms
the sum of the exponents of the variables in a monomial
Monomial
a polynomial with one term
Binomial
a polynomial with two terms
6 terms
Dividing Polynomials
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
26 terms
Polynomials
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
Polynomials
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
15 terms
Finding Zeros and Their Multiplicities
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)
Polynomials
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
Unit 5: Polynomials & Polynomial Functions
Binomial Theorem
Conjugates
Degree of a Polynomial
Difference of Cubes
Binomials of the form a + √b and a - √b.
The greatest of the degrees of its terms after it has been si…
Binomial Theorem
Conjugates
Binomials of the form a + √b and a - √b.
20 terms
Polynomials
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.
22 terms
Polynomials
monomial
binomial
trinomial
polynomial
an expression with only one term
an expression with two terms
an expression with three terms
a monomial or the sum or difference of monomials
monomial
an expression with only one term
binomial
an expression with two terms
Polynomials 1
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
Operations with Polynomials & Factoring
Polynomial
Binomial
Trinomial
Degree of a Monomial
A monomial or sum of monomials
The sum of two monomials
The sum of three monomials
The sum of the exponents of the variables.
Polynomial
A monomial or sum of monomials
Binomial
The sum of two monomials
Unit 2 Polynomials
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
25 terms
The Language of Polynomials
Poly
Nomial
Coefficient
Linear Equation
Many
Term
Number in front of the variable
y = mx + b
Poly
Many
Nomial
Term
11 terms
polynomials
LINEAR POLYNOMIAL
QUADRATIC POLYNOMIAL
CUBIC POLYNOMIAL
ZEROES OF LINEAR POLYNOMIAL
a polynomial of degree 1
a polynomial of degree 2
a polynomial of degree 3
alpha
LINEAR POLYNOMIAL
a polynomial of degree 1
QUADRATIC POLYNOMIAL
a polynomial of degree 2
14 terms
Solving Polynomials Graphically
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.
14 terms
3.1 Solving Polynomials Graphically
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.
14 terms
Unit 7 - Classifying Polynomials
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…
Polynomials
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.
26 terms
Polynomials
Polynomial
Monomial
Binomial
Trinomial
many terms together that can include operations, variables, e…
a polynomial with 1 term
a polynomial with 2 terms
a polynomial with 3 terms
Polynomial
many terms together that can include operations, variables, e…
Monomial
a polynomial with 1 term
16 terms
Polynomials
Monomial
Binomial
Polynomial
Long Division with Polynomials
a polynomial with one term
a polynomial with two terms, the sum of two monomials
consisting of many monomials
Monomial
a polynomial with one term
Binomial
a polynomial with two terms, the sum of two monomials
17 terms
Polynomials
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
17 terms
Polynomials
polynomial
degree of a polynomial
standard form of a polynomial
monomial
monomial or sum of monomials
highest exponent
in order from highest exponent to lowest exponent
only 1 term
polynomial
monomial or sum of monomials
degree of a polynomial
highest exponent
Exponents & Polynomials
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
61 terms
precalc H - unit 02 - Quadratics & Polynomials
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
12 terms
Degrees of Polynomials
1
2
5
4
degree of 7x
degree of 10x²
degree of 4wx³y + 3x²y
degree of 2x³y
1
degree of 7x
2
degree of 10x²
10 terms
The Rational Zeros Theorem
Find all possible rational roots for... x…
Find all possible rational roots for... x…
Find all possible rational roots for... x…
Find all possible rational roots for... x…
Find all possible rational roots for... x…
Find all possible rational roots for... x…
Dividing Polynomials
(7x² + 14x)÷(x + 2)
(90x + x³ + 20x² - 100) ÷ (x + 10)
(x³ - 5x² - 30x - 54) ÷ (x - 9)
(7x² + 36x - 36) ÷ (x + 6)
7x
x²+10x-10
x²+4x+6
7x-6
(7x² + 14x)÷(x + 2)
7x
(90x + x³ + 20x² - 100) ÷ (x + 10)
x²+10x-10
27 terms
Polynomials
monomial
binomial
trinomial
polynomial
an expression with only one term
an expression with two terms
an expression with three terms
a monomial or the sum or difference of monomials where each m…
monomial
an expression with only one term
binomial
an expression with two terms
13 terms
Polynomials
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…
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