Rational Zeros of Function

Question

find all the rational zeros of these functions

x³ - 10x² + 23x - 14

x³ + 6x² + 9x + 4

Answer 1

We will use the Rational Root Theorem, or p/q test, for both problems.

Given: q*x^n + ... + p

Possible Rational Roots will be ±[factors of p / factors of q]

We will then use synthetic division to see if they are roots.

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Problem #1:

x³ - 10x² + 23x - 14

p = 14

q = 1

p/q = 14/1

Possible Rational Roots will be ±[(14 * 1, 7 * 2) / 1] = ±{1,2,7,14} = 1, -1, 2, -2, 7, -7, 14, -14

Synthetic Division #1:

1 ⌋ 1 -10 23 -14

1 -9 14

-------------------------

1 -9 14 ⌉ 0

Therefore, x = 1 is a root

Synthetic Division #2:

7 ⌋ 1 -9 14

7 -14

-------------------------

1 -2 ⌉ 0

Therefore, x = 7 is a root

Synthetic Division #3:

2 ⌋ 1 -2

2

-------------------------

1 ⌉ 0

Therefore, x = 2 is a root

The Rational Roots for problem #1 are x = 1, 2, 7

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Problem #2:

x³ + 6x² + 9x + 4

p = 4

q = 1

p/q = 4/1

Possible Rational Roots will be ±[(4 * 1, 2 * 2) / 1] = ±{1,2,4} = 1, -1, 2, -2, 4, -4

Synthetic Division #1:

-1 ⌋ 1 6 9 4

-1 -5 -4

-------------------------

1 5 4 ⌉ 0

Therefore, x = -1 is a root

Synthetic Division #2:

-4 ⌋ 1 5 4

-4 -4

-------------------------

1 1 ⌉ 0

Therefore, x = -4 is a root

Synthetic Division #3:

-1 ⌋ 1 1

-1

-------------------------

1 ⌉ 0

Therefore, x = -1 is a root

The Rational Roots for problem #2 are x = -1, -1, -4

NOTE: -1 is a double root