JeopardyLabs

Remainders

100

Explain what a remainder is.

A part, number, or quantity that is left over. 

200

Explain the Remainder Theorem.

f(x)=(x-c)*q(x)+r(x), f(c)=(c-c)*q(c)+r, therefore, f(c)=r

300

Use the remainder theorem to find the remainder when 3x³-2x²+x-5 is divided by x+2

(x+2) so f(2),

3(-2)³-2(-2)²+(-2)-5

-24-8-2-5

R= -39

400

Divide (2x³-7x²+4x-5) by (x-3)

2x²-x+1 R -2

500

Divide 2x⁴-3x³+9x²-14x+7 by x²+4

2x²-3x+1 R -2x+3

Polynomials

100

What is a polynomial?

An expression consisting of multiple terms

200

What is the quadratic formula?

(-b(+/-)sqrt(b²)-4(a)(c))/(2a)

300

Solve the polynomial inequality 2x³+13x²-8x<52 graphically

(-6.5, -2)U(2,infinity)

400

Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicity include 2(multiplicity of 2), 3+i

x⁴-10x³+38x²-64x+40

500

Using the given zero of 1+3i, find all the zeros of f(x)=x⁴-2x³+5x²+10x-50

Synthetically divide by 1+3i, synthetically divide by 1-3i

x³-5x²

x(x²-5)

x=+/- 5

Factors

100

Describe what a factor of a polynomial is.

Breaking up polynomials into simpler terms

200

Explain the Factor Theorem.

When f(c)=0 then x-c is a factor of f(x)

300

Factor f(x)=x⁴-7x²+12

f(x)=(x²-4)(x²-3)

f(x)=(x+2)(x-2)(x²-3)

400

Use the Factor Theorem to determine whether x+3 is a factor of x³+2x²-4x-2

f(-3)=1

so not a factor

500

Write f(x)=x³-x²-x-2 as a product of linear and irreducible quadratic factors with all real coefficients.

(x-2)(x²+x+1)

Division

100

Name two types of division

Long division and synthetic division

200

Divide 99 by 11

9

300

Divide 3x³–x–7 by x–2 using synthetic division

3x²+6x+11 R 15

400

Find all zeros and write a linear factorization of the function f(x)=x³+x+10

synthetically divide by -2

x²-2x+5

quadratic formula

x=1(+/-)2i

(x+2)(x-1-2i)(x-1+2i)

500

Using the given zero, final all of the zeros and write a linear factorization of the function. 1+i is a zero of f(x)=x⁴-2x³-x²+6x+6

Synthetically divide using 1+i

Synthetically divide using 1-i

x²-3=0

x=sqrt(3), -sqrt(3)

(x-1-i)(x-1+i)(x+sqrt(3))(x-sqrt(3))

Zeros

100

If (x-2) is a factor of a polynomial, name one possible zero

x=2

200

If (x+8) is a factor of a polynomial, what is a possible zero?

x=-8

300

If (x-1+3i) is a factor of a polynomial, name two possible zeros

x=1-3i and x=1+3i

400

What are the potential rational zeros of f(x)=2x⁴-x³-4x²-x-6

+/-1, +/-1/2, +/-2, +/-3, +/-3/2, +/-6

500

Find all the real zeros of the function. Identify each zero as rational or irrational. f(x)=x⁴-7x²+12

x=+/- 2 and x=+/- sqrt(3) or 3^(1/2)