Problem 41 - A Graphical Approach to College Algebra

Question

Without graphing, answer true or false to each statement. Then, support your answer by graphing. The function \(f(x)=-3 x^{4}+5\) has two real zeros.

Step-by-Step Solution

Answer

True. The function \(f(x) = -3x^4 + 5\) has two real zeros.

Step 1: Understand the Function

The given function is a polynomial: \( f(x) = -3x^4 + 5 \). It is a quartic polynomial (degree 4), and since the leading coefficient is negative, the graph opens downward.

Step 2: Determine Zeros of the Function

To find the zeros of the function, we set \( f(x) = 0 \): \( -3x^4 + 5 = 0 \). Solve for \( x^4 \): \( x^4 = \frac{5}{3} \).

Step 3: Analyze Real Solutions

Consider \( x^4 = \frac{5}{3} \). Since \( x^4 = (x^2)^2 \), and any non-negative real number has a real square root, \( x^2 = \sqrt{\frac{5}{3}} \) or \( x^2 = -\sqrt{\frac{5}{3}} \). Only \( x^2 = \sqrt{\frac{5}{3}} \) is valid, giving two real roots \( x = \pm \sqrt[4]{\frac{5}{3}} \).

Step 4: Count the Real Zeros

There are two real zeros \( x = \sqrt[4]{\frac{5}{3}} \) and \( x = -\sqrt[4]{\frac{5}{3}} \), hence the statement is true.

Step 5: Graph to Support Answer

If graphed, \( f(x) = -3x^4 + 5 \) will intersect the x-axis at two points, proving that there are indeed two real zeros.

Key Concepts

Real Zeros

Real zeros are the values of \( x \) that make the polynomial equal to zero. For a function like \( f(x) = -3x^4 + 5 \), finding real zeros means solving \( -3x^4 + 5 = 0 \). This will tell us at which points the graph intersects the x-axis.

• Set the polynomial equal to zero: \( -3x^4 + 5 = 0 \)
• Solve for \( x^4 \): \( x^4 = \frac{5}{3} \)

To find real zeros, simplify further by taking the fourth root:

• Find \( x \) using \( x = \pm \sqrt[4]{\frac{5}{3}} \)
• This results in two real zeros: One positive \( (\sqrt[4]{\frac{5}{3}}) \) and one negative \( (-\sqrt[4]{\frac{5}{3}}) \)

Real zeros are important because they indicate where the graph of the function touches or crosses the x-axis.

Quartic Polynomial

A quartic polynomial is a polynomial of degree four. The highest power of \( x \) is 4 in such polynomials. For example, \( f(x) = -3x^4 + 5 \) is a quartic polynomial.

• Quartic polynomials can have up to four real zeros, but they might have fewer if some are complex.
• The sign of the leading coefficient affects the direction the arms of the graph face. A negative leading coefficient, as in our example, makes the graph open downwards.
• The general shape of a quartic graph can vary widely, having up to three turning points.

Quartic polynomials like \( -3x^4 + 5 \) are symmetrical around the y-axis when all terms involve even powers of \( x \). Recognizing these characteristics helps predict the graph's behavior.

Graphing Techniques

Understanding graphing techniques can make visualizing quartic polynomials much simpler. For \( f(x) = -3x^4 + 5 \), graphing helps confirm the existence and placement of real zeros.

• Identifying Intercepts: The y-intercept occurs where \( x = 0 \). Here, \( f(0) = 5 \).
• End Behavior: Because the leading coefficient is negative, as \( x \to \pm \infty \), \( f(x) \to -\infty \). The arms of the graph face downward.
• Symmetry: This function is even, meaning it's symmetrical about the y-axis.

Graphing visually supports the algebraic solution and confirms that the real zeros found were accurate. Technologies like graphing calculators or software tools can generate these graphs quickly, giving immediate visual confirmation.