Problem 115 - Algebra and Trigonometry

Question

Exercises 113–115 will help you prepare for the material covered in the next section. Use $$ \frac{2 x^{3}-3 x^{2}-11 x+6}{x-3}=2 x^{2}+3 x-2 $$ to factor \(2 x^{3}-3 x^{2}-11 x+6\) completely

Step-by-Step Solution

Answer

The completely factored form of \(2x^3 - 3x^2 - 11x + 6\) is \((x-3)(2x - 1)(x + 2)\).

Step 1: Dividing the polynomial

Based on the given equation, perform the division \((2x^3 - 3x^2 - 11x + 6) / (x-3)\). You will obtain \(2x^2 + 3x - 2\), the quadratic term on the right hand side of the equation, thus confirming that the division is correct.

Step 2: Rewrite the polynomial as a product

Now, rewrite the polynomial \(2x^3 - 3x^2 - 11x + 6\) as a product of \((x - 3)\) and \((2x^2 + 3x - 2)\), which results in \((x-3)(2x^2 + 3x - 2)\). This confirms that the polynomial has been factored.

Step 3: Factoring the quadratic function

Next, factor the quadratic function \(2x^2 + 3x - 2) \). This can be done by looking for two numbers that when multiplied give -4 (2*-2) and when added give 3. Those two numbers are -1 and 4. Therefore, \((2x^2 + 3x - 2)\) factors to \((2x - 1)(x + 2)\).

Step 4: Final Factoring

Finally, substitute \((2x - 1)(x + 2)\) into \((x-3)(2x^2 + 3x - 2)\) from Step 2. The fully factored form of the given polynomial is then \((x-3)(2x - 1)(x + 2)\).

Key Concepts

Synthetic Division

Synthetic division is a quick and efficient method to divide polynomials when the divisor is a linear factor of the form \(x-a\). It simplifies the process by using only the coefficients of the polynomial, rather than the full terms, making it less cumbersome than long division.

Let's break it down:

• Start with the polynomial you wish to divide and the divisor in the form \(x - a\).
• Use the root, \(a\), derived from the divisor \((x-a)\) as the number you will divide by in synthetic division.
• Set up the coefficients in a row, drop down the first coefficient to the row below, and multiply it with \(a\).
• Add the product to the next coefficient, and repeat the process until you're done.

The results will give you the coefficients of the quotient polynomial and any remainder. In the example \(\frac{2x^3 - 3x^2 - 11x + 6}{x-3}\), synthetic division confirms that the quotient is \(2x^2 + 3x - 2\).

This method is particularly useful when verifying factorization, as it confirms whether a polynomial is divisible by \(x-a\) without leaving a remainder.

Quadratic Factorization

Quadratic factorization involves breaking down a quadratic equation into simpler multiplicative components. For any quadratic trinomial of the form \(ax^2 + bx + c\), factoring allows us to express it as a product of two binomials.

To factor a quadratic, such as \(2x^2 + 3x - 2\), one can follow these steps:

• Identify a pair of numbers that multiply to give the product of \(a\) and \(c\) (in this case \(2 \times -2 = -4\)), and add to give \(b\) (\(3\)).
• Decompose the middle term using the numbers found (here, \(-1\) and \(4\)).
• Group the terms into two pairs and factor by grouping.
• Finally, write the expression as a product of two binomials using common factors (achieved solution: \((2x - 1)(x + 2)\)).

Quadratic factorization simplifies solving quadratic equations, finding intercepts, or further simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operational symbols, forming math sentences that describe relationships or problems.

These expressions can be manipulated and transformed through various techniques. Here's how they relate to our example problem:

• Simplifying: Using division (as seen in synthetic division), expressions are reduced to their simplest form.
• Factoring: The expression was rewritten into a product of simpler expressions \((x-3)(2x - 1)(x + 2)\). This makes it easier to solve for roots or understand the expression's structure.
• Expanding: The opposite of factoring; it's useful for double-checking by multiplying factors back out to arrive at the original expression.

Manipulating algebraic expressions through factoring and division not only aids in simplifying them but also unveils their characteristics, such as roots and potential asymptotes in graphs. Mastering these concepts is crucial for solving complex equations and proving identities in algebra.