Problem 43 - 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 \(f(x)\). $$ g(x)=x+5, \quad f(x)=9 x^{3}+21 x^{2}-104 x+80 $$

Step-by-Step Solution

Answer

The complete factorization of \(f(x)\) is \((x+5)(3x-4)^2\).

Step 1: Set Up Synthetic Division

To perform synthetic division, first identify the zero of the divisor \(g(x) = x + 5\). Set \(x + 5 = 0\) to find \(x = -5\). Next, write the coefficients of \(f(x) = 9x^3 + 21x^2 - 104x + 80\), which are: \(9, 21, -104, 80\). Place \(-5\) outside the synthetic division bracket and the coefficients inside.

Step 2: Perform Synthetic Division

Begin by bringing down the leading coefficient \(9\) to the bottom row. Multiply \(-5\) by \(9\) to get \(-45\), and add it to \(21\) to get \(-24\). Continue the process: Multiply \(-5\) by \(-24\) to get \(120\), add to \(-104\) to get \(16\). Finally, multiply \(-5\) by \(16\) to get \(-80\), add to \(80\) to get \(0\). The final remainder is \(0\), confirming \(x + 5\) is a factor.

Step 3: Interpret the Result

The synthetic division provides the quotient polynomial which is \(9x^2 - 24x + 16\) since the remainder is 0. This means \(f(x)\) can be expressed as the product of \(g(x) = x+5\) and the quotient \(9x^2 - 24x + 16\).

Step 4: Factor the Quadratic Quotient

Factor the quadratic \(9x^2 - 24x + 16\) using the discriminant method or by grouping. By factoring, it simplifies to \((3x - 4)(3x - 4)\) or \((3x - 4)^2\).

Step 5: Write the Complete Factorization

Combine all the factors to write the complete factorization of the polynomial \(f(x)\). Since \(f(x) = (x+5)(3x-4)^2\), this represents the complete factorization of \(f(x) = 9x^3 + 21x^2 - 104x + 80\).

Key Concepts

Polynomial Factorization

Polynomial factorization is like deconstructing a mathematical expression to its basic building blocks, similar to breaking down composite numbers into prime factors. This process involves expressing a polynomial as a product of its simpler, non-trivial polynomial factors. For example, consider the polynomial function \(f(x) = 9x^3 + 21x^2 - 104x + 80\). To factorize it, we seek to write it in terms of smaller polynomial components. This can reveal roots or simplify a problem we're working with.

The factorization process usually involves identifying factors like linear functions (e.g., \(x + 5\)) and quadratic functions (e.g., \(3x^2 - 5x + 2\)).

• Linear factors show immediate roots of the polynomial.
• Quadratic factors are further reduced into simpler binomials.

Finding these factors often uses methods such as synthetic division, which efficiently tests potential roots to simplify the polynomial further.

Quadratic Factorization

Quadratic factorization is a specific type of polynomial factorization. When you encounter a quadratic polynomial, like \(9x^2 - 24x + 16\), the goal is to express it as the product of two binomials: \((ax + b)(cx + d)\).

To factor a quadratic:

• Check if there are obvious factors or roots.
• Use methods like the quadratic formula, completing the square, or factor by grouping.

Sometimes, a given quadratic can be factored by inspection, especially if it simplifies to a perfect square. For example, \(9x^2 - 24x + 16\) simplifies to \((3x - 4)^2\), an indication that it's a perfect square trinomial. Recognizing these patterns speeds up solving and simplifies factorizations.

Polynomial Division

Polynomial division allows you to divide one polynomial by another and find a quotient and remainder, akin to dividing two numbers in arithmetic. One specialized and efficient method for dividing polynomials is synthetic division. It significantly reduces computational complexity, especially useful for linear divisors.

Let's put this into practice with an example:

• To divide \(f(x) = 9x^3 + 21x^2 - 104x + 80\) by \(g(x) = x + 5\), first identify the zero: \(x = -5\).
• Then perform synthetic division using \(-5\) and the coefficients of \(f(x)\) (which are 9, 21, -104, 80).
• You systematically calculate each step as outlined in the process, resulting in understanding if \(g(x)\) is a factor.

This division not only tests for factors but also gives you a clear way to rephrase the original polynomial into a quotient (simplified polynomial) plus any remainder.

Remainder Theorem

The Remainder Theorem is a very useful concept in algebra that relates to polynomial division. It states that when a polynomial \(f(x)\) is divided by a linear divisor of the form \(x - c\), the remainder of this division is simply \(f(c)\).

This concept is advantageous because:

• It provides a quick method to find the remainder without having to perform synthetic division or long division fully.
• If \(f(c) = 0\), this means \(x - c\) is a factor of \(f(x)\), indicating \(c\) is a root of the polynomial.

In the problem given, the successful use of synthetic division resulting in a remainder of zero confirms \(x + 5\) as a factor. Consequently, according to the Remainder Theorem, evaluating the polynomial \(f(x)\) at \(-5\) would also confirm there is no remainder, ensuring a smooth collaboration between algebraic principles and procedural operations like synthetic division.