Pre-Calculus Zeros

Rational Zero Theorem

Descartes' Rule

Steps to Finding Zeros

Factoring

Actual Zeros using RZT

100

List all possible rational zeros.

f(x) = x³+ 11x²- 15x - 27

+/- 1, +/- 3, +/- 9, +/- 27

100

Give possible positive and negative real zeros.

f(x) = x³+ 11x²- 15x - 27

Positive: 1

Negative: 2 or 0

100

List all possible rational zeros.

f(x) = x³- 19x+ 30

+/- 1, +/- 2, +/- 3, +/- 5, +/- 6, +/- 10, +/- 15, +/- 30,

100

Find zeros by factoring

f(x) = 2x³+ 5x²- 24x - 60

x = {-5/2, +/- 2√ 3}

100

List all possible rational zeros. Then, find the actual zeros using synthetic and factoring.

f(x) = x³+ 2x²- 11x - 12

x = {-4, -1, 3}

200

List all possible rational zeros.

f(x) = x³+ 87x²- 16x - 45

+/- 1, +/- 5, +/- 9, +/- 45

200

Give possible positive and negative real zeros.

f(x) = 3x⁴- x³- 63x²- 39x + 20

Positive: 2 or 0

Negative: 2 or 0

200

Use synthetic division to divide the polynomial by 3

f(x) = x³- 19x+ 30

x² - 3x - 10 +(60/(x-3))

200

Find zeros by factoring

f(x) = x⁴- 9x²+ 18

x = {+/- √ 3, +/- √ 6}

200

List all possible rational zeros. Then, find the actual zeros using synthetic and factoring.

f(x) = x³- x²- 25x +25

x = {-5, 1, 5}

300

List all possible rational zeros.

f(x) = 2x³- 5x²+ 8x - 20

+/- 1, +/- 2, +/- 4, +/- 5, +/- 10, +/- 20, +/- 1/2, +/- 5/2

300

Give possible positive and negative real zeros.

f(x) = 2x⁴- x³- 2x²+ x

Positive: 2 or 0

Negative: 1

300

Use synthetic division to divide the polynomial by -5

f(x) = x³- 19x+ 30

x² - 5x + 6

300

Find zeros by factoring

f(x) = 2x³ - 54

x = {3, (-3+/-3i√ 3)/2}

300

List all possible rational zeros. Then, find the actual zeros using synthetic and factoring.

f(x) = 2x³+ 7x²- 17x - 10

x = {-5, -1/2, 2}

400

List all possible rational zeros.

f(x) = 4x³- 7x²+ 2x + 1

+/- 1, +/- 1/2, +/- 1/4

400

Give possible positive and negative real zeros.

f(x) = 9x⁵ - 3x⁴+ 10x³- x²+ 27x - 9

Positive: 5 or 3 or 1

Negative: 0

400

Knowing that -5 is a zero of the following polynomial

f(x) = x³- 19x+ 30

and when you divide by it by -5, you end up with the polynomial

x² - 5x + 6

Give the complete factorization of the problem.

f(x) = (x + 5)(x - 3)(x - 2)

400

Find zeros by factoring

f(x) = x⁴ - 81

x = {+/- 3, +/- 3i}

400

List all possible rational zeros. Then, find the actual zeros using synthetic and factoring.

f(x) = x⁴ + x³- 31x²- 61x - 30

x = {-5, -1, 6}

500

List all possible rational zeros.

f(x) = 3x⁴- x³- 63x²- 39x + 20

+/- 1, +/- 2, +/- 4, +/- 5, +/- 10, +/- 20, +/- 1/3, +/- 2/3, +/- 4/3, +/- 5/3, +/- 10/3, +/- 20/3

500

Give possible positive and negative real zeros.

f(x) = 7x⁶ - 12x⁵ + 15x⁴ - 8x³ - x² + 2x - 1

Positive: 5 or 3 or 0

Negative: 1

500

Knowing the complete factorization of a polynomial is

f(x) = (x + 5)(x - 3)(x - 2)

find the zeros

x = {-5, 2, 3}

500

Find zeros by factoring and use to write complete factorization of the function.

f(x) = 4x⁴- 33x²+ 8

f(x) = (2x + 1)(2x - 1)(x + 2√ 2)(x - 2√ 2)

500

List all possible rational zeros. Then, find the actual zeros using synthetic and factoring.

f(x) = 6x⁴ - 7x³- 9x²+ 7x + 3

x = {-1, -1/3, 1, 3/2}