Problem 27 - College Algebra

Question

Find each product. $$(2 x-5)(7 x+2)$$

Step-by-Step Solution

Answer

\[14x^2 -31x -10\]

Step 1: Distribute the First Term

Multiply the first term in the first binomial (2x) with each term in the second binomial. This gives: \[2x * 7x + 2x * 2\], which simplifies to \[14x^2 + 4x\]

Step 2: Distribute the Second Term

Multiply the second term in the first binomial (-5) with each term in the second binomial. This gives: \[-5 * 7x -5 * 2\], which simplifies to \[-35x -10\]

Step 3: Combine Like Terms

Combine the results from steps 1 and 2: \[14x^2 + 4x - 35x -10\]. Simplify to get the final answer. Combine the x terms: \[14x^2 -31x -10\]

Key Concepts

Binomials

Binomials are a type of polynomial with exactly two terms. They can be expressed in the form \( ax + b \), where \( a \) and \( b \) are constants, and \( x \) is a variable. In the given exercise, the binomials are \( (2x - 5) \) and \( (7x + 2) \). When dealing with binomials, each term is important because you'll often perform operations like addition, subtraction, or multiplication with them.
Understanding the structure of binomials helps when you start multiplying them using methods like distribution or FOIL (First, Outer, Inner, Last).
Binomials help simplify expressions and are fundamental in calculus and algebra. By mastering binomials, students can solve complex equations and work with quadratic functions more easily.

Distribution Method

The distribution method is crucial when multiplying binomials. Often known as the distributive property of multiplication, it states that \( a(b + c) = ab + ac \). It ensures that each term in one binomial is multiplied by each term in the other.
In the problem provided, the distribution method is used to multiply \( (2x - 5) \) by \( (7x + 2) \).

• Step 1 involved multiplying the first term \( 2x \) in the first binomial by the terms \( 7x \) and \( 2 \) in the second binomial to get \( 14x^2 + 4x \).
• Step 2 involved multiplying the second term \( -5 \) in the first binomial by \( 7x \) and \( 2 \) to get \( -35x - 10 \).

This method breaks down complex multiplication into more manageable parts, making it easier to see how each component contributes to the final solution.

Combining Like Terms

Combining like terms is often the final step in simplifying polynomials. It involves adding or subtracting terms that have the same variable raised to the same power.
After distributing the terms in each binomial, as seen in the exercise above, you get the expression \( 14x^2 + 4x - 35x - 10 \). The like terms here are \( 4x \) and \( -35x \). By combining these, you get \( -31x \).
This is essential because combining like terms simplifies equations, making them easier to understand and solve. It also helps in finding the most reduced form of a polynomial, which is often key in further algebraic operations or in mathematical problem-solving contexts.