Problem 44 - Algebra for College Students

Question

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+4, \quad f(x)=4 x^{3}+4 x^{2}-39 x+36 $$

Step-by-Step Solution

Answer

Yes, \(x + 4\) is a factor, and \(f(x) = (x + 4)(2x - 3)^2\).

Step 1: Set up Synthetic Division

To perform synthetic division, we first set up the coefficients of the polynomial \(f(x) = 4x^3 + 4x^2 - 39x + 36\). These coefficients are \([4, 4, -39, 36]\). The divisor \(g(x) = x + 4\) corresponds to \(x = -4\). We will use \(-4\) for the synthetic division setup.

Step 2: Perform Synthetic Division

Start by bringing down the first coefficient, 4. Multiply \(-4\) by 4 and add to the next coefficient: \(4 \cdot (-4) = -16\), and \(4 + (-16) = -12\). Repeat this process: \((-4)(-12) = 48\), and \(-39 + 48 = 9\). Then, \((-4)(9) = -36\), and \(36 + (-36) = 0\). The remainder is 0, confirming \(x + 4\) is a factor.

Step 3: Write the Divided Polynomial

The result of the synthetic division yields the coefficients \(4, -12, 9\), which correspond to the polynomial \(4x^2 - 12x + 9\). Since the remainder is 0, \(g(x) = x + 4\) is a factor, and \(f(x)\) can be expressed as \((x + 4)(4x^2 - 12x + 9)\).

Step 4: Factor the Quadratic

The quadratic \(4x^2 - 12x + 9\) can be factored further. Looking for factors of \(4 \times 9 = 36\) that sum to \(-12\), we find \(-6\) and \(-6\). Thus, \(4x^2 - 12x + 9 = (2x - 3)(2x - 3)\), or \((2x - 3)^2\).

Step 5: Complete the Factorization

Combining the factors from synthetic division and the factorization of the quadratic, we find: \(f(x) = (x + 4)(2x - 3)^2\), which completes the factorization of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is similar to dividing numbers but applies to polynomials. It allows us to divide a polynomial by another polynomial of lesser degree. There are two primary methods: long division and synthetic division.

• Long Division: Works like the division of numbers, aligning terms in descending degree and dividing step by step, aligning with coefficients.
• Synthetic Division: A shortcut specifically for dividing by linear polynomials like \(x + c\). This method uses only coefficients and is efficient when the divisor is linear.

For the synthetic division of \(f(x) = 4x^3 + 4x^2 - 39x + 36\) by \(g(x) = x + 4\), we transform \(g(x)\) to \(x = -4\) and follow these steps for division.
Synthetic division is particularly useful because it reduces the complexity and time required to divide polynomials compared to long division.

Factorization

Factorization involves breaking down a complex expression into a product of simpler factors. This is like splitting a number into its prime factors but applies to polynomials.
In our example, we want to factor \(f(x)\). To start, synthetic division revealed that \(x + 4\) is a factor. This means \(f(x)\) can be written as \((x + 4)(4x^2 - 12x + 9)\).
Next, we factor the quadratic polynomial \(4x^2 - 12x + 9\). By finding the values that multiply to 36 and add to -12, we identified it could be expressed as \((2x - 3)^2\). This complete factorization shows \(f(x)\) as \((x + 4)(2x - 3)^2\).
Through factorization, we convert the polynomial into a multiplication of its factors, revealing insights into its roots and simplifying the expression for further analysis.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two, generally in the form \(ax^2 + bx + c\). These polynomials can be factored or solved using several techniques: factoring, completing the square, or using the quadratic formula.
In the exercise, the quadratic \(4x^2 - 12x + 9\) results from the initial polynomial division. Efficient factoring involved finding two identical factors, \( (2x - 3)(2x - 3) \), indicating a perfect square trinomial.
Recognizing and factoring quadratics aids in understanding more profound characteristics, like finding roots or intersections, and is a crucial step in comprehensively solving algebra problems.

Remainder Theorem

The Remainder Theorem provides a simple way to evaluate polynomials when they are divided by linear factors. If the polynomial \(f(x)\) is divided by \(x-c\), the remainder of this division is the same as evaluating \(f(c)\).
In our problem, using synthetic division with \(-4\) (derived from divisor \(x + 4\)), the remainder was 0. When the remainder is 0, \(x+4\) is a factor of \(f(x)\). This application confirms the polynomial divides evenly.
Utilizing the Remainder Theorem simplifies the verification of factors, aiding students and mathematicians alike in quickly understanding the nature of polynomial expressions and their behaviors.