One of the many factors of the polynomial function y=10x^4 - 7 x^3 - 34 x^2 + 37 x - 6 is in the form of (ax+b), where a and b are real numbers such that a < b. Determine the value of (8a - 3b).

Options: 

25

-8

-12

4

43

2

11None of the given options are correct.


To determine one of the linear factors \((ax + b)\) of the polynomial \(y = 10x^4 - 7x^3 - 34x^2 + 37x - 6\), we can use the Rational Root Theorem. The theorem states that any rational root of the polynomial, expressed in simplest form as \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term (\(-6\)) and \(q\) is a factor of the leading coefficient (\(10\)).

The factors of \(-6\) are \(\pm 1, \pm 2, \pm 3, \pm 6\).

The factors of \(10\) are \(\pm 1, \pm 2, \pm 5, \pm 10\).

Therefore, the possible rational roots are:

\[
\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{3}{5}, \pm \frac{6}{5}, \pm \frac{1}{10}, \pm \frac{3}{10}, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{3}{5}, \pm \frac{6}{5}
\]

We will test these possible rational roots by substituting them into the polynomial \(y = 10x^4 - 7x^3 - 34x^2 + 37x - 6\).

Let's test \(x = 1\):

\[10(1)^4 - 7(1)^3 - 34(1)^2 + 37(1) - 6 = 10 - 7 - 34 + 37 -

Explore how to find a linear factor of the polynomial function y = 10x^4 - 7x^3 - 34x^2 + 37x - 6 using the Rational Root Theorem. This problem involves identifying rational roots and testing them with the polynomial equation. Suitable for high school students.

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