#### 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

3 real zeros

4 real zeros

3 real zeros (one double zero)

one real zero

Sketch the graph and identify the number of real zeros:... f(x)…

Sketch the graph and identify the number of real zeros:... f(x)…

Sketch the graph and identify the number of real zeros:... y = x…

Sketch the graph and identify the number of real zeros:... f(x)…

3 real zeros

Sketch the graph and identify the number of real zeros:... f(x)…

4 real zeros

Sketch the graph and identify the number of real zeros:... f(x)…

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

Write the polynomial in factored form.

What are the zeros of the function? Gr…

What are the zeros of the function? Wh…

what is the relative maximum and minim…

4x(x-4)(x+6)

0,-2,-5

The number 0 is a zero multiplicity 2; the numbers 1 and 3 ar…

The relative maximum is at (-1.53, 12.01) and the relative mi…

Write the polynomial in factored form.

4x(x-4)(x+6)

What are the zeros of the function? Gr…

0,-2,-5

f(x)=(x+2)(x-2i)(x+2i)

f(x)=(x+2)^2(x+i)(x-i)

f(x)=(x+2)(x-5+i)(x-5-i)

f(x)=(x+2)(x-i)(x+i)(x+5+i)(x+5-i)

Degree of 3 and zeros -2 and 2i

Degree of 4 and zeros -2 with multiplicity of 2 and -i

Degree of 3 and zeros -2 and 5-i

Degree of 5 and zeros -2, i, and -5-i

f(x)=(x+2)(x-2i)(x+2i)

Degree of 3 and zeros -2 and 2i

f(x)=(x+2)^2(x+i)(x-i)

Degree of 4 and zeros -2 with multiplicity of 2 and -i

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…

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)

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)

Leading Coefficient: -... Degree: Even... #…

Leading Coefficient: -... Degree: Even... #…

Leading Coefficient: +... Degree: Odd... # o…

Leading Coefficient: +... Degree: Even... #…

Leading Coefficient: -... Degree: Even... #…

Leading Coefficient: -... Degree: Even... #…

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 +…