Problem 60 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(5 x-2)\left(6 x^{2}+2 x-1\right)$$

Step-by-Step Solution

Answer

The product is \(30x^3 - 2x^2 - 9x + 2\).

Step 1: Distribute the First Term

Begin by distributing the first term of the first binomial, \(5x\), to each term in the second polynomial \((6x^2 + 2x - 1)\). This gives:\[5x \cdot 6x^2 + 5x \cdot 2x + 5x \cdot (-1) = 30x^3 + 10x^2 - 5x\]

Step 2: Distribute the Second Term

Next, distribute the second term of the first binomial, \(-2\), to each term in the second polynomial \((6x^2 + 2x - 1)\). This results in:\[-2 \cdot 6x^2 + (-2) \cdot 2x + (-2) \cdot (-1) = -12x^2 - 4x + 2\]

Step 3: Combine Like Terms

Now, we combine the products obtained in steps 1 and 2:\[30x^3 + 10x^2 - 5x - 12x^2 - 4x + 2\]Simplify by combining like terms:- Combine the \(x^2\) terms: \(10x^2 - 12x^2 = -2x^2\)- Combine the \(x\) terms: \(-5x - 4x = -9x\)This results in the polynomial:\[30x^3 - 2x^2 - 9x + 2\]

Step 4: Write the Final Expression

The final polynomial expression after combining like terms is:\[30x^3 - 2x^2 - 9x + 2\]

Key Concepts

Binomials

A binomial is a simple algebraic expression that contains exactly two terms. These terms are typically separated by a plus or minus sign. For instance, in the problem

• (5x - 2) and
• (6x^2 + 2x - 1)

both include binomials.
Understanding how to multiply these kinds of expressions is essential in algebra.
Working with binomials helps students grasp more complex math concepts as it forms the foundation for polynomial algebra.
While binomials may look simple, their multiplication requires a few steps to make sure all parts are accounted for, using properties like distribution.

Distributive Property

The distributive property is a key concept in algebra that involves multiplying a single term by each term within a binomial or polynomial. This technique is crucial when dealing with expressions like \((5x - 2)(6x^{2} + 2x - 1)\).
The distributive property states that for any numbers or variables, a(b + c) = ab + ac.
In our example:

• First, multiply each term in a(5x)aby each term in the second polynomial (6x^2 + 2x - 1).
• Next, repeat the process with the second term a(-2).

This ensures every term is accounted for, and none is left out.
Using the distributive property correctly is crucial for simplifying expressions and obtaining accurate results.

Combining Like Terms

Once all terms have been distributed using the distributive property, the next step in simplifying an expression is combining like terms.
Like terms are those that have the same variable raised to the same power, such as \(10x^2\) and \(-12x^2\).
To combine like terms, you add or subtract their coefficients:

• Combine (\(x^2\)) terms like \(10x^2 - 12x^2\) resulting in \(-2x^2\).
• Combine (\(x\)) terms like \(-5x - 4x\)in the final expression, yielding \(-9x\).

Combining like terms simplifies the polynomial and is a pivotal step in ensuring that your final answer is as clear and concise as possible.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the crux of algebra and can represent a wide variety of mathematical problems. In this exercise, we dealt with expressions such as \(5x - 2\) and \(6x^2 + 2x - 1\).
These expressions can be manipulated using multiplication, division, addition, and subtraction to solve problems or simplify calculations.
Understanding how to handle algebraic expressions, especially in the context of polynomial multiplication, is vital:

• It helps in applying properties like distribution effectively.
• It enhances problem-solving skills by simplifying complex equations into manageable expressions.
• It is foundational for advanced studies in algebra, calculus, and beyond.

By mastering these expressions, students can confidently tackle a wide range of mathematical challenges.