Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Find all real zeros of the function. $$f(x)=x^{4}-11 x^{2}+18$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=x^{4}-11 x^{2}+18\) are \(\sqrt{2}, -\sqrt{2}, 3, -3\).

Step 1: Transform polynomial

Rewrite the function in terms of \(x^2\), replacing \(x^4\) with \((x^2)^2\), to get a quadratic equation in the form of \(a u^2 + bu + c = 0\), where \(u = x^2\). This yields \(u^2 - 11u + 18 = 0\).

Step 2: Solve quadratic equation

Solve for \(u\) using the quadratic formula, with \(a=1\), \(b=-11\), and \(c=18\). This results in \(u = [11 \pm \sqrt{{(-11)^2 - 4*1*18}}] / 2*1\), which simplifies to \(u = [11 \pm \sqrt{{121 - 72}}] / 2 = [11 \pm \sqrt{{49}}] / 2\). Thus, \(u = 2, 9\).

Step 3: Find real values of x

Remembering that \(u = x^2\), substitute \(2\) and \(9\) back in for \(u\) to find the values of \(x\). This leads to \(x = \sqrt{2}, -\sqrt{2}, \sqrt{9}, -\sqrt{9}\), which simplifies to \(x = \sqrt{2}, -\sqrt{2}, 3, -3\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that are set equal to zero and include a variable raised to the second power. They are usually written in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are constants and \( a \eq 0 \). The solutions to quadratic equations, which are the values of the variable that make the equation true, are called 'roots' or 'zeros'. These can be found using several methods such as factoring, completing the square, or using the quadratic formula.

For example, in the function \( f(x) = x^4 - 11x^2 + 18 \), a substitution is used to transform the fourth-degree polynomial into a quadratic equation by letting \( u = x^2 \). This reduces the problem to finding the zeros of \( u^2 - 11u + 18 = 0 \), a more manageable quadratic equation.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as the product of its factors, which are polynomials of a lower degree. When dealing with quadratic polynomials, factoring can be a straightforward method of finding the real zeros without resorting to more complex solutions.

The factoring process usually starts with identifying a common factor in all terms or applying techniques like grouping. For quadratics such as \( ax^2 + bx + c \), we seek two binomials \( (dx + e)(fx + g) \), whose product equals the original quadratic equation. However, not all quadratic equations are factorable over the integers. In such cases, we might still factor over the real numbers or use other methods to find the zeros. For example, the equation \( u^2 - 11u + 18 = 0 \) from the given exercise can be factored as \( (u - 2)(u - 9) = 0 \), where the solutions for \( u \) are clearly seen as \( 2 \) and \( 9 \).

Quadratic Formula

The quadratic formula is a universally applicable method to find the zeros of any quadratic equation, given that it's in the form \( ax^2 + bx + c = 0 \). The formula is \( x = [-b \pm \sqrt{b^2 - 4ac}]/(2a) \). The expression under the square root, \( b^2 - 4ac \), is called the 'discriminant', and it determines the nature of the roots, which can be real and distinct, real and equal, or complex.

In the context of the exercise, after using the substitution \( u = x^2 \) to obtain \( u^2 - 11u + 18 = 0 \), we would apply the quadratic formula with \( a=1 \: b=-11 \: c=18 \), which simplifies to \( u = [11 \pm \sqrt{(11)^2 - 4 \times 1 \times 18}]/2 \). The narrowed down solutions for \( u \) are calculated as \( 2 \) and \( 9 \), after solving the quadratic equation. These values are then further substituted back into the original variable \( x \) to find the real zeros of the function: \( \sqrt{2} \), \( -\sqrt{2} \), \( 3 \) and \( -3 \).