Problem 36 - 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 \(f(x)\). $$ g(x)=x+1, \quad f(x)=x^{3}+6 x^{2}-31 x-36 $$

Step-by-Step Solution

Answer

The complete factorization is \((x+1)(x+9)(x-4)\).

Step 1: Identify Roots in Synthetic Division

For using synthetic division to check if \(g(x) = x+1\) is a factor of \(f(x) = x^3 + 6x^2 - 31x - 36\), we identify the root as \(x = -1\).

Step 2: Setup Synthetic Division

Using \(x = -1\), set up the synthetic division. Write the coefficients of \(f(x)\) in a row: \([1, 6, -31, -36]\).

Step 3: Perform Synthetic Division

Carry down the first coefficient, \(1\). Multiply this by \(-1\) and add to the next coefficient: \(6-1 = 5\). Repeat this process: \(-5\) added to \(-31\) gives \(-36\), and continue with \(36-36\) resulting in \(0\). The final row of coefficients is \([1, 5, -36, 0]\).

Step 4: Factorization Interpretation

The remainder is \(0\) indicating \(x+1\) is a factor. The quotient coefficients \([1, 5, -36]\) represent the quadratic factor \(x^2 + 5x - 36\).

Step 5: Quadratic Factorization

Factor \(x^2 + 5x - 36\) by finding two numbers that multiply to \(-36\) and add to \(5\). The numbers \(9\) and \(-4\) work. Thus, \(x^2 + 5x - 36 = (x+9)(x-4)\).

Step 6: Complete the Factorization

The complete factorization of \(f(x)\) is \(f(x) = (x+1)(x+9)(x-4)\) based on the synthetic division result and the quadratic factorization.

Key Concepts

Factorization

Factorization is a method used to break down a mathematical expression into its simplest form by expressing it as a product of its factors. It helps in simplifying complex algebraic expressions and solving equations. In the context of polynomial functions, factorization involves writing a polynomial as a product of its linear or lower-degree factors.
For example, given a polynomial function like \(f(x) = x^3 + 6x^2 - 31x - 36\), factorization allows us to express it as \((x+1)(x+9)(x-4)\), facilitating easier computation in solving equations or analyzing the function's behavior.
Understanding factorization is crucial because it provides insight into the roots of the polynomial and helps in simplifying problems in algebra by breaking them into manageable parts.

Polynomial Division

Polynomial division is a mathematical process that involves dividing a polynomial by another polynomial, similar to how long division is performed with numbers. This method includes long division and synthetic division, each with its applications.

• Synthetic Division: This is a simplified form of polynomial division used primarily when dividing by a linear polynomial, such as \(x-c\). It streamlines calculations and is particularly useful for testing potential factors quickly. For example, using synthetic division, you can verify if \(x+1\) is a factor of the polynomial \(x^3 + 6x^2 - 31x - 36\) by testing if \(x = -1\) results in a remainder of zero.

Synthetic division is especially handy in zeroing in on the roots of polynomials, as it allows you to efficiently determine factors, which simplify further operations and manipulations of the expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition and subtraction) that represent mathematical relationships. They form the foundation of algebraic problem-solving and are crucial in representing real-world problems mathematically.
For instance, the expression \(x^3 + 6x^2 - 31x - 36\) represents a polynomial expression involving a variable \(x\). By performing operations like factorization, we can gain insights into its behavior and solve for unknown quantities.
Understanding and manipulating these expressions are key to solving equations, modeling scenarios, and constructing functions. Mastering algebraic expressions is pivotal for progressing to more advanced mathematical topics.

Quadratic Equations

Quadratic equations are a specific type of polynomial equation that have the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Solving quadratic equations often involves finding the values of \(x\) that make the equation true.
One common method of solving these equations is by factorization. For example, the quadratic resulting from synthetic division \(x^2 + 5x - 36\) can be factored into \((x+9)(x-4)\). Setting each factor equal to zero shows that the roots are \(x = -9\) and \(x = 4\).
Understanding quadratic equations is essential because they frequently appear in various branches of mathematics and science. They model many natural phenomena and underpin the principles of calculus and other advanced disciplines.