Question

PLEASE HELP ME

5) Let g(x) = 12x3 – 20x2 + x + 3

5A) Apply the Rational Roots Theorem to list all possible rational roots of g:


3: 1,3
12: 1, 2, 3, 4, 6, 12


5B) Use synthetic division to show that x = -1/3 is a zero of g:

Answer 1

see pic

Explanation:

5A) You're part way there! You also need to divide each factor of 3 by each factor of 12. Then simplify and remove all duplicates. All of the possible roots will be +/- also. (See pic)

5B) see pic

Answer 2

A: ±{1/12, 1/6, 1/4, 1/3, 1/2, 3/4, 1, 3/2, 3}

B: quotient from division by 3x+1: 4x² -8x +3; remainder: 0

Explanation:

You want the possible rational roots of g(x) = 12x³ -20x² +x +3, and you want to use synthetic division to show that -1/3 is a zero.

5A) Rational roots theorem

The rational roots theorem tells you that possible rational roots will be of the form ...

±{divisor of the constant}/{divisor of the leading coefficient}

You have listed the divisors of each. The possible rational roots are the unique rational numbers that result:

±{1/12, 1/6, 1/4, 1/3, 1/2, 3/4, 1, 3/2, 3}

5B) Synthetic division

To show that -1/3 is a zero, we can divide g(x) by (x +1/3) and look for a remainder of 0.

The synthetic division calculator we use to do this (first attachment) treats this as division by (3x+1), effectively, 3(x +1/3). The division by 3 is done first, so the dividend becomes 4x³ -20/3x² +1/3x +1. Then the synthetic division of that by x+1/3 is carried out.

The quotient is 4x² -8x +3, and the remainder is zero. The zero remainder shows that -1/3 is a zero of g(x).

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Additional comment

The quadratic factor of g(x), namely 4x² -8x +3, can be further factored as ...

(4x -2)(4x -6)/4 = (2x -1)(2x -3)

This shows the other zeros of g(x) are 1/2 and 3/2, values on the list in part A. This is confirmed by the graph in the second attachment.