Problem 84 - Algebra and Trigonometry Real Mathematics, Real People

Question

Match the cubic function with the correct number of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ;\) Irrational zeros: \(\mathbf{0}\) (c) Rational zeros: \(1 ; \quad\) Irrational zeros: 2 (d) Rational zeros: \(1 ;\) Irrational zeros: \(\mathbf{0}\) $$f(x)=x^{3}-2 x$$

Step-by-Step Solution

Answer

The correct match for the function \(f(x) = x^{3}-2x\) is (c) which shows one Rational zero and two Irrational zeros.

Step 1: Factorise the Polynomial

The given polynomial can be factored as follows: \(f(x)=x^{3}-2x = x(x^{2}-2)\).

Step 2: Solve for Real Roots

The factorised equation yields the roots as follows: \(x(x^{2}-2) = 0\). This gives us 3 roots, \(x = 0, \sqrt{2}, -\sqrt{2}\). Note that \(\sqrt{2}\) and \(-\sqrt{2}\) are irrational numbers.

Step 3: Classify Roots

Based on the roots found, \(x = 0\) is a rational root (integer) (only one) and \(x = \sqrt{2}, -\sqrt{2}\) are irrational roots (making two). Therefore, the correct choice matching the polynomial should have one rational root and two irrational roots.

Key Concepts

Rational Zeros

Rational zeros are solutions to a polynomial equation that can be expressed as a ratio of two integers, such as \( \frac{a}{b} \). These solutions are key steps in solving polynomial equations and can provide insight into the behavior of a function.

• A rational zero is a number like 0, 2, or \( \frac{1}{3} \) that fits into a polynomial equation without resulting in a non-repeating decimal or irrational number.
• These zeros are found using the Rational Root Theorem, which states that any rational solution of a polynomial is a factor of the polynomial’s constant term divided by a factor of the leading coefficient.
• In the cubic function \( f(x) = x^3 - 2x \), the rational zero is \( x = 0 \). This is because 0 is an integer, fitting the criteria for a rational number.

Recognizing rational zeros helps in simplifying the polynomial for further analysis.

Irrational Zeros

Irrational zeros are solutions to a polynomial that can't be written as a simple fraction. They often involve square roots that cannot be simplified to a rational number.

• These zeros can't be expressed as exact decimals; they appear as non-repeating, non-terminating decimals.
• Irrational zeros often come in conjugate pairs, like \( \sqrt{2} \) and \( -\sqrt{2} \), as seen in the polynomial \( f(x) = x^3 - 2x \).
• One must usually use approximation methods or radical expressions to work with irrational zeros effectively.

Thus, irrational roots require more careful handling in computations but reveal important characteristics about the polynomial.

Factoring Polynomials

Factoring a polynomial breaks it down into simpler "factors" that multiply together to get the original polynomial. This process is useful because it allows us to find the zeros of the polynomial more easily.

• Factoring often starts by finding the greatest common factor or using patterns like difference of squares.
• In the given function \( f(x) = x^3 - 2x \), factoring simplifies it to \( x(x^2 - 2) \), revealing its roots.
• This makes it evident that the polynomial can be solved by setting each factor equal to zero, hence unraveling its zeros: \( x = 0, \sqrt{2}, -\sqrt{2} \).

Mastering polynomial factoring not only simplifies solving equations but also aids in analyzing the behavior and characteristics of the polynomial function itself.