Problem 13 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ 2 x^{3}-3 x^{2}-4 x+6 $$

Step-by-Step Solution

Answer

The expression factors to \((x^2 - 2)(2x - 3)\).

Step 1: Group and Factor by Pair

First, we need to group the expression into two pairs: \[ (2x^3 - 3x^2) + (-4x + 6) \]Now, factor each group separately. In the first group, \(2x^3 - 3x^2\), we can factor out \(x^2\): \[ x^2(2x - 3) \]In the second group, \(-4x + 6\), we can factor out \(-2\): \[ -2(2x - 3) \]

Step 2: Identify Common Binomial Factor

Notice that both terms have a common binomial factor \(2x - 3\): The expression now can be reorganized as \[ x^2(2x - 3) - 2(2x - 3) \]This allows us to factor out \((2x - 3)\) from the entire expression.

Step 3: Factor Out the Common Binomial

Factor \((2x - 3)\) out from both terms to write the expression as a product of binomials: \[ (x^2 - 2)(2x - 3) \]

Step 4: Verify the Result

Expand \((x^2 - 2)(2x - 3)\) to verify our factorization:\[ x^2(2x) + x^2(-3) - 2(2x) - 2(-3) \]Simplify to get:\[ 2x^3 - 3x^2 - 4x + 6 \]Since the result is the original polynomial, our factorization is correct.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They serve as a foundation for solving various mathematical problems. In this exercise, you are given an algebraic expression involving variables, specifically related to a polynomial:

• The expression is: \( 2x^3 - 3x^2 - 4x + 6 \)
• Each term in the expression is composed of coefficients (like 2, -3, and -4) and variables raised to powers (like \(x^3, x^2\)).
• The different powers of the variable \(x\) show that this is a polynomial expression.

Algebraic expressions are flexible, which means you can manipulate them using mathematical rules to solve equations or simplify problems. Understanding the structure and form of these expressions is crucial for correctly factoring or expanding them in algebra.

Binomial Factoring

Binomial factoring is a method used to break down complex polynomials into simpler, product-of-binomial expressions. This can make polynomials easier to work with, especially when solving equations or simplifying them.

• The given polynomial \( 2x^3 - 3x^2 - 4x + 6 \) is factored as the product of two binomials: \((x^2 - 2)(2x - 3)\).
• The process begins by grouping terms and looking for common factors in pairs of terms.
• By isolating \((2x - 3)\) as a common binomial factor, you're actually exploiting a key concept in factoring. This makes solving or manipulating the equation more straightforward.

Factoring binomials involves recognizing patterns and manipulating algebraic expressions effectively, which is a valuable skill in various areas of algebra.

Polynomial Expansion

Polynomial expansion is essentially reversing the process of factoring. It's done by multiplying factors to return to the original full expression. This helps verify the correctness of the factorization.

• When you've factored an expression into smaller parts (binomials in this case), you can always expand it back.
• In the expansion of \( (x^2 - 2)(2x - 3) \), each term in the first binomial is multiplied by each term in the second binomial.
• The result, \( 2x^3 - 3x^2 - 4x + 6 \), should match the original expression if the factorization was done correctly.

Expansion allows for a quick verification of your work. Mastering both factoring and expansion techniques allows for fluency in managing polynomials, offering reliable ways to simplify, solve, and interpret algebraic expressions.