Problem 18 - Precalculus with Calculus Previews

Question

Use synthetic division to determine whether the indicated linear polynomial is a factor of the given polynomial function \(f\). If yes, find all other zeros and then give the complete factorization of \(f(x)\) $$ x-2 ; f(x)=x^{3}-6 x^{2}-16 x+48 $$

Step-by-Step Solution

Answer

Yes. The complete factorization is \((x-2)(x-6)(x+4)\). Zeros are \(x=2, 6, -4\).

Step 1: Set up Synthetic Division

To use synthetic division, we rewrite the divisor \( x-2 \) in the form of \( x-c \). Here, \( c = 2 \). Write the coefficients of \( f(x) = x^3 - 6x^2 - 16x + 48 \) as a row, so we have: 1, -6, -16, 48. Write 2 to the left side.

Step 2: Perform Synthetic Division

Bring down the leading coefficient 1 into the bottom row. Multiply 2 (the value from \( x-c \)) by 1 and write the result (2) under the next coefficient. Add the column: \( -6 + 2 = -4 \). Repeat: Multiply 2 by -4 to get -8, write under -16, add to get -24. Multiply 2 by -24 to get -48, write under 48, add to get 0.

Step 3: Interpret Remainder

The final bottom row shows the coefficients of the quotient polynomial and the remainder. The remainder is 0, which means \( x-2 \) is a factor. Coefficients are for \( x^2 - 4x - 24 \). If remainder is zero, \( x - c \) is a factor.

Step 4: Factor the Quotient Polynomial

Further factorize \( x^2 - 4x - 24 \). Find two numbers multiplying to -24 (constant term) and adding to -4 (linear term). These are 2 and -12. Rewrite: \( x^2 - 4x - 24 = (x-6)(x+4) \).

Step 5: Find Remaining Zeros

Set each factor from \((x-6)(x+4)\) equal to zero: \(x-6=0\) and \(x+4=0\). Solving these gives \(x=6\) and \(x=-4\). Thus, zeros are \( x = 2, 6, -4 \).

Step 6: Write Complete Factorization

Since \( x-2 \) is a factor and we have found \((x-6)(x+4)\), the complete factorization of \( f(x) \) is \((x-2)(x-6)(x+4)\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in a single variable is \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\]. The degree of the polynomial is the highest power of the variable, which dictates the "complexity" in terms of curves and intersections with the x-axis.

Key characteristics of polynomial functions include:

• Continuity: Polynomial functions are smooth and continuous.
• Coefficients: The numbers multiplying the powers of the variable, showing strength or contribution of each part.
• Roots or Zeros: Points where the polynomial evaluates to zero, important for solving polynomials.

Factorization

Factorization is breaking down a polynomial into simpler "factors" that when multiplied together give the original polynomial. It is a crucial step in understanding and solving polynomial equations.

• Factorization allows us to find roots or zeros of the polynomial, which are solutions to the equation.
• For quadratic polynomials, factorization is straightforward and involves finding two values that sum to the middle coefficient and multiply to the constant term.
  
  
• For higher-degree polynomials, techniques like synthetic division and the Rational Root Theorem help in finding factors efficiently.
  
  

Factorization not only helps in solving polynomials but also aids in simplifying complex expression, making calculations more manageable.

Zeros of a Polynomial

The zeros of a polynomial (also called roots or solutions) are the values of the variable that make the polynomial equal zero. Finding these zeros is significant because they tell us where the graph of the polynomial will intersect the x-axis.

To find the zeros:

• Use techniques like factoring, synthetic division, or setting the polynomial equal to zero to solve for the variable.
• For the polynomial \(f(x) = x^3 - 6x^2 - 16x + 48\), we found zeros using synthetic division and further factorization, resulting in zeros at \(x = 2, 6,\) and \(-4\).

Recognizing zeros not only aids in graph analysis but also in further mathematical modeling and problem-solving.

Synthetic Division Process

The synthetic division process is a simplified form of dividing a polynomial by a binomial of the form \(x - c\). It's especially useful because it's quicker and often more intuitive than long division.

Steps for Synthetic Division:

• Setup: Write down the coefficients of the polynomial and the constant \(c\) from \(x - c\) in a systematic layout.
• Begin Calculation: Bring down the first number (leading coefficient). Multiply this number by \(c\) and add it to the next coefficient. Repeat this process for all coefficients.
• Interpreting Results: The last number in the row is the remainder. If it's zero, \(x-c\) is a factor of the polynomial. The other numbers form the coefficients of the quotient polynomial.

An example of synthetic division in action helps to confirm whether a binomial is a factor and thereby further understand the polynomial's structure and roots.