Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Match the function with its exact number of zeros. $$f(x)=x^{3}+3 x^{2}+2 x$$ (a) 1 zero (b) 3 zeros (c) 4 zeros (d) 5 zeros

Step-by-Step Solution

Answer

(b) 3 zeros

Step 1: Identify the degree of the polynomial

The degree of a polynomial corresponds to the highest power in the expression. In the given function \(f(x)=x^{3}+3x^{2}+2x\), it is clear that the highest power (and thus, the degree) is 3.

Step 2: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every polynomial of degree 'n' has exactly 'n' complex zeros, counting multiplicities. Since the given polynomial is of degree 3, it is understood that it must have exactly 3 zeros.

Step 3: Compare with given options

Since the number of zeros is determined to be 3, option (b) which says '3 zeros' is the correct answer.

Key Concepts

Zeros of Polynomials

In algebra, zeros of a polynomial function are the values of the variable that make the polynomial equal to zero. These are also known as the roots of the equation. A zero can be:

• Real or complex numbers. Real zeros are the ones you can actually see on the graph as points where the curve crosses the x-axis. Complex zeros, however, may not always be seen on a typical graph because they have imaginary components.
• Single or multiple occurrences, often referred to as the 'multiplicity' of the zero. When a zero repeats, such as \((x - a)^2 = 0\) yielding \(x = a\) twice, it is said to have a multiplicity of two.

Like finding a solution to an equation, finding the zeros helps understand the nature of the polynomial function. Hence, to find the precise number of zeros of the function \(f(x)=x^{3}+3x^{2}+2x\), you set \(f(x) = 0\) and solve for \(x\). Since this function is a degree 3 polynomial, it should have exactly 3 zeros according to the Fundamental Theorem of Algebra.

Degree of a Polynomial

The degree of a polynomial is a term you often encounter in algebra, which indicates the highest power of the variable in the polynomial equation. In the function \(f(x)=x^{3}+3x^{2}+2x\), it is clear that the highest power of \(x\) is 3. Therefore, this polynomial is a degree 3 polynomial.

• Leading Term: The term with the highest power, for this polynomial, is \(x^3\).
• Importance of the Degree: The degree not only tells us about the polynomial's overall behavior but also gives us important information about the number of solutions or zeros we can expect.

This number, the degree, also gives us a clue to the maximum number of turns the graph of the polynomial might take and how it behaves at the ends of the graph. Remember, higher-degree polynomials can have highly intricate graphs!

Fundamental Theorem of Algebra

Understanding the Fundamental Theorem of Algebra is crucial to solving polynomial problems. This theorem states that every non-zero polynomial equation with complex coefficients has a number of zeros (including multiplicity) equal to its degree.

• Significance of the Theorem: This means that a polynomial of degree \(n\) will have \(n\) solutions in the complex number system, which includes real numbers.
• Application: In the case of our polynomial, \(f(x)=x^{3}+3x^{2}+2x\), because it is degree 3, it will have exactly 3 zeros.

The theorem assures us that solutions exist for polynomials, pointing to a finite and predictable number of zeros based on the polynomial's degree. This understanding aids in graphing and analyzing polynomial functions. Therefore, it helps to assure when and why polynomials have specific numbers of solutions, which is pivotal for solving algebraic equations.