According to the Rational Root Theorem, which could be a factor of the polynomial f(x) = 6x4 – 21x3 – 4x2 + 24x – 35?

2 answers:

3 0

Answer: (2x – 7) (3x^3 – 2x – 5)

SOLVINGS
Given the polynomial f(x) = 6x^4 – 21x^3 – 4x^2 + 24x – 35

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case,
The Leading Coefficient is 6
The Trailing Constant is -35.

Factors of the Leading Coefficient are 1, 2, 3 and 6
Factors of the Trailing Constant are 1, 5, 7, and 35

Testing for the rational roots,
If P= 7 and Q = 2
P/Q = 3.5
F (P/Q) = 0.00

Using the Factor Theorem; which states that if P/Q is root of a polynomial then this polynomial can be divided by Q.x – PTherefore, the polynomial 6x^4 – 21x^3 – 4x^2 + 24x – 35 is divisible by 2x –
7

Factorizing 2x – 7
Divide the polynomial into two groups (6x^4 – 21x^3 and – 4x^2 + 24x – 35)

Factorizing Group 1
6x^4 – 21x^3 divided by 2x – 7 = 3x^3
∴ 6x^4 – 21x^3 = 3x^3 (2x – 7) ….. (Group 1)

Factorizing Group 2
– 4x^2 + 24x – 35 divided by 2x – 7 = -2x+5
∴ 4x^2 + 24x – 35 = (-2x+5)(2x – 7) ….. (Group 2)

Bringing together Groups 1 and 2
6x^4 – 21x^3 – 4x^2 + 24x – 35 = (2x – 7) (3x^3 – 2x – 5)