Problem 34 - Algebra and Trigonometry

Question

Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. $$ P(x)=x^{4}-2 x^{3}+8 x-16 $$

Step-by-Step Solution

Answer

The polynomial factors to \((x - 2)(x + 2)(x^2 - 2x + 4)\) with real zeros at \(x = 2\) and \(x = -2\). The graph crosses the x-axis only at these points and rises on both ends.

Step 1: Identify Possible Factoring Techniques

To factor the polynomial \(P(x) = x^4 - 2x^3 + 8x - 16\), identify any common factoring techniques. For this expression, it seems beneficial to try grouping terms to factor the polynomial.

Step 2: Group Terms

Group the first two terms and the last two terms: \((x^4 - 2x^3)\) and \((8x - 16)\). This allows us to look for common factors in these pairs.

Step 3: Factor Each Group

In the first group \((x^4 - 2x^3)\), factor out \(x^3\) to get \(x^3(x - 2)\). For the second group \((8x - 16)\), factor out 8, giving \(8(x - 2)\). The polynomial is now expressed as \(x^3(x-2) + 8(x-2)\).

Step 4: Factor Out Common Binomial

Notice that \((x - 2)\) is common in both terms. Factor \((x - 2)\) out, giving the expression \((x-2)(x^3 + 8)\).

Step 5: Further Factor \(x^3 + 8\) Using Sum of Cubes

Recognize that \(x^3 + 8\) can be factored using the sum of cubes formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(a = x\) and \(b = 2\). Thus, \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\).

Step 6: Write the Fully Factored Form

The fully factored form of the polynomial is \((x - 2)(x + 2)(x^2 - 2x + 4)\).

Step 7: Find the Zeros of the Polynomial

Set each factor equal to zero: \(x - 2 = 0\) gives \(x = 2\), and \(x + 2 = 0\) gives \(x = -2\). The quadratic \(x^2 - 2x + 4 = 0\) has no real roots (discriminant is negative). Therefore, the only real zeros are \(x = 2\) and \(x = -2\).

Step 8: Sketch the Graph

Plot the zeros \(x = 2\) and \(x = -2\) on the x-axis. Since \(x^2 - 2x + 4\) has no real roots, the graph will intercept the x-axis only at these points. Additionally, because the leading term is \(x^4\), the graph will rise on both ends. Sketch the graph accordingly.

Key Concepts

Zeros of Polynomials

To understand zeros of polynomials, imagine them as points where the graph of the polynomial hits the x-axis. These are the solutions of the equation when the polynomial is set to zero.

For the polynomial \( P(x) = x^4 - 2x^3 + 8x - 16 \), we discover its zeros by factoring it. After factoring, we get the expression \((x - 2)(x + 2)(x^2 - 2x + 4)\). To find the zeros, we set each factor equal to zero.

• \( x - 2 = 0 \) gives \( x = 2 \)
• \( x + 2 = 0 \) gives \( x = -2 \)

The quadratic \( x^2 - 2x + 4 = 0 \) does not have real solutions because its discriminant is negative. Thus, the real zeros of \( P(x) \) are \( x = 2 \) and \( x = -2 \). These zeros help determine where the polynomial graph touches or crosses the x-axis. Finding zeros is essential for graphing and solving polynomial equations, making them a fundamental concept in algebra.

Sum of Cubes Formula

The sum of cubes is a special formula used in algebra to factor expressions of the form \( a^3 + b^3 \). The formula is given by:
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]

In the problem, we used it to factor \( x^3 + 8 \). This expression is recognized as a sum of cubes because \( x^3 \) is \( a^3 \) and \( 8 = 2^3 \) is \( b^3 \). Applying the formula here:

• \( a = x \)
• \( b = 2 \)

So, \( x^3 + 8 \) becomes:
\[(x + 2)(x^2 - 2x + 4)\]
This formula is very powerful because it provides a straightforward way to break down and simplify expressions that seem complex at first. By understanding and applying the sum of cubes formula, we can factor polynomials that might otherwise be tricky to handle, thus revealing more about their structure and solutions.

Graphing Polynomials

Graphing polynomials involves plotting their zeros and understanding their end behavior. For the polynomial \( P(x) = x^4 - 2x^3 + 8x - 16 \), the factored form \((x - 2)(x + 2)(x^2 - 2x + 4)\) gives valuable insights.

We found that the real zeros are \( x = 2 \) and \( x = -2 \). These zeros are the x-intercepts of the graph. The polynomial does not intercept at any other point because \( x^2 - 2x + 4 \) yields no real roots. Knowing this limits where the graph touches the x-axis.

• The degree of \( P(x) \) is 4 (the highest power of x), indicating that the ends of the graph will point in the same direction.
• The leading coefficient is positive, which means both ends will rise as they extend away from the zeros.

To sketch the graph, mark the zeros on the x-axis and draw the curve so it touches these points. Then sketch the rising arms according to the end behavior determined by the leading term. Understanding these steps helps visualize how polynomial functions behave across the plane.