Question

According to the Rational Root Theorem, which is not a possible zero of the function \(f(x)=40 x^5-42 x^4-107 x^3+107 x^2+33 x-36 ?\) (A) \(-\frac{2}{3}\) (B) \(-\frac{3}{8}\) (C) \(\frac{3}{4}\) (D) \(\frac{4}{5}\)

Short answer

\(-\frac{3}{8}\) is not a possible zero of the function.

Step-by-step solution

Identify the constant term and the leading coefficient

The constant term of the polynomial function \(f(x)=40 x^5-42 x^4-107 x^3+107 x^2+33 x-36\) is -36 and the leading coefficient is 40.

Find the factors of the constant term and the leading coefficient

The factors of -36 are \(-1, 1, -2, 2, -3, 3, -4, 4, -6, 6, -9, 9, -12, 12, -18, 18, -36, 36\). The factors of 40 are \(-1, 1, -2, 2, -4, 4, -5, 5, -8, 8, -10, 10, -20, 20, -40, 40\). Note that in the Rational Root Theorem, it's implied that a possible rational root can take any sign, so we include the negatives.

Form the possible ratios

The possible ratios consist of every factor of -36 divided by every factor of 40.

Compare the possible ratios to the given options

From the list of possible ratios, you can discern that \(-\frac{2}{3}\), \(\frac{3}{4}\), and \(\frac{4}{5}\) are possible zeros of the function. However, \(-\frac{3}{8}\) is not on the list, so it is not a possible zero of the function.

Key concepts

Polynomials

Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. A polynomial can have many degrees, depending on the highest power of the variable. This degree dictates the behavior and shape of the polynomial's graph. For instance, a polynomial of degree 5, like the one in the given problem, can produce five distinct curves or "turns".

Each term in a polynomial is usually ordered from highest to lowest degree, making it easier to observe the leading coefficient and constant term. For the function defined as \(f(x) = 40x^5 - 42x^4 - 107x^3 + 107x^2 + 33x - 36\), the leading coefficient is the coefficient of the highest degree term, which is 40, and the constant term is the standalone number, -36. This constant often helps determine possible rational roots.

Zero of a Function

Zeros of a function, also known as roots, are the input values \(x\) for which the function \(f(x)\) evaluates to zero. In simple terms, these zeros are the \(x\)-coordinates at which the graph of the polynomial intersects the x-axis.

For a polynomial, these zeros can be real or complex. When it comes to real roots, they can be further classified as rational or irrational. Rational roots are those that can be expressed as a fraction of two integers. Identifying these rational roots becomes crucial when solving polynomial equations since they simplify the problem-solving process.

Factors

Factors are numbers that divide another number without leaving a remainder. In the context of the Rational Root Theorem, we find the factors of both the constant term and the leading coefficient of the polynomial. This helps in determining which rational numbers could potentially be zeros of the polynomial.

For the polynomial \(f(x) = 40x^5 - 42x^4 - 107x^3 + 107x^2 + 33x - 36\), the factors of the constant term (-36) include \(-1, 1, -2, 2, -3, 3\), among others. Likewise, the factors of the leading coefficient (40) are \(-1, 1, -2, 2, 4, -4, 5\), etc. These factors are then paired to form possible rational zeros.

Rational Roots

Finding rational roots is made convenient with the Rational Root Theorem. This mathematical principle suggests that any rational root of a polynomial equation is in the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

Following this theorem, one can list all possible combinations of \(\frac{p}{q}\) to find potential rational zeros. By comparing these possibilities against the given options, we can quickly discern which numbers could be actual roots. For example, since \(-\frac{3}{8}\) is not one of these combinations, it is determined not to be a possible zero, making it the correct answer in the exercise context.