Question

1. solve \\(3x^5 + 15x = 18x^3\\) for \\(x\\). which of the following is not a factor?

3

\\(-\sqrt{5}\\)

\\(\sqrt{5}\\)

1

Explanation:

Rearrange the polynomial equation

Using the Algebraic Simplification knowledge point

$$
LATEXBLOCK0
$$

Factor out the greatest common factor

Using the Factoring Polynomials knowledge point

$$
LATEXBLOCK1
$$

Factor the quadratic in form

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK2
$$

Fully factor the expression

We factor the remaining quadratic terms over the real numbers:

• \(x^2 - 1 = (x - 1)(x + 1)\)
• \(x^2 - 5 = (x - \sqrt{5})(x + \sqrt{5})\)

Combining all factors, the fully factored equation is:

$$ 3x(x - \sqrt{5})(x + \sqrt{5})(x - 1)(x + 1) = 0 $$

Identify the non-factor

We compare the given options with our list of factors:

• \(3\) is a factor (the constant coefficient of the monomial \(3x\)).
• \(-\sqrt{5}\) is not a factor. The factor is the binomial \((x + \sqrt{5})\), whereas \(-\sqrt{5}\) is a root (or zero) of the polynomial, not a polynomial factor itself.
• \(\sqrt{5}\) is not a factor for the same reason (it is a root, corresponding to the factor \(x - \sqrt{5}\)).
• \(1\) is a trivial factor of any expression, but looking at the options, we must distinguish between algebraic factors of the polynomial and its roots.

Let's re-evaluate standard high school terminology. Often, questions of this type confuse "factors" with "roots/solutions" or refer to the linear factors of the form \((x - c)\) where \(c\) is a root.
If the question asks "Solve... Which of the following is NOT a factor?", and the options are numbers:
The roots of the equation are:

$$ x = 0, \quad x = \sqrt{5}, \quad x = -\sqrt{5}, \quad x = 1, \quad x = -1 $$

Among the choices:

• \(-\sqrt{5}\) is a root (corresponding to factor \(x + \sqrt{5}\)).
• \(\sqrt{5}\) is a root (corresponding to factor \(x - \sqrt{5}\)).
• \(1\) is a root (corresponding to factor \(x - 1\)).
• \(3\) is a coefficient, but \(3\) is NOT a root of the equation since \(3(3)^5 + 15(3) = 729 + 45 = 774

eq 18(27) = 486\).

Thus, the question uses "factor" to mean "a value of \(x\) that satisfies the equation" (i.e., a root/solution, which is a value that makes the linear term \((x - c)\) a factor). Under this standard curriculum phrasing, the roots are factors of the polynomial's root set, meaning \(3\) is NOT a solution/root.

Answer:

• (A) 3 (Correct answer)
• (B) -\(\sqrt{5}\)
• (C) \(\sqrt{5}\)
• (D) 1