Problem 84 - Algebra A Combined Function

Question

Perform the indicated operations. \((2 x-1)(10 x-7)\)

Step-by-Step Solution

Answer

The simplified expression is: \(20x^2 - 24x + 7\).

Step 1: Apply the Distributive Property

To simplify the expression \((2x - 1)(10x - 7)\), first apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last). This involves multiplying each term in the first binomial by each term in the second binomial.

Step 2: Multiply the First Terms

Multiply the first terms of each binomial: \(2x\) and \(10x\). This gives: \(2x \times 10x = 20x^2\).

Step 3: Multiply the Outer Terms

Multiply the outer terms: \(2x\) and \(-7\). This gives: \(2x \times -7 = -14x\).

Step 4: Multiply the Inner Terms

Multiply the inner terms: \(-1\) and \(10x\). This gives: \(-1 \times 10x = -10x\).

Step 5: Multiply the Last Terms

Multiply the last terms: \(-1\) and \(-7\). This gives: \(-1 \times -7 = 7\).

Step 6: Combine the Products

Now, combine all the products from the previous steps: \(20x^2\), \(-14x\), \(-10x\), and \(7\).

Step 7: Simplify the Expression

Combine the like terms, which are the \(-14x\) and \(-10x\) terms: \(-14x - 10x = -24x\). Thus, the simplified expression is: \(20x^2 - 24x + 7\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, helping you simplify expressions involving multiplication over addition or subtraction. It's the principle that allows you to multiply a single term by each term within a parenthesis.
When operating on the expression \((2x - 1)(10x - 7)\), you'll use each term in the first binomial to multiply every term in the second binomial. This strategy ensures that all possible products are calculated.
In essence, the distributive property expands the expression to include each pairwise multiplication, creating a new expression that is a sum or difference of these products. This property is key to handling polynomial multiplication effectively.

FOIL Method

The FOIL method is a specific application of the distributive property used predominantly for multiplying two binomials. It's an acronym for First, Outer, Inner, and Last, reflecting the order in which you multiply terms. Here's a closer look at the method:

• First: Multiply the first terms in each binomial. In our example, this is \(2x \times 10x = 20x^2\).
• Outer: Multiply the outer terms, which are the first term of the first binomial and the last term of the second: \(2x \times -7 = -14x\).
• Inner: Multiply the inner terms, the last term of the first binomial and the first term of the second: \(-1 \times 10x = -10x\).
• Last: Multiply the last terms in each binomial: \(-1 \times -7 = 7\).

Using the FOIL method helps to ensure no terms are missed during multiplication. The products are then combined through addition or subtraction, leading to the final expanded form.

Binomial Expressions

Binomial expressions are algebraic expressions that consist of two terms separated by a plus or minus sign. For example, \(2x - 1\) and \(10x - 7\) are binomials. These expressions are the building blocks for more complex polynomials.
Each term in a binomial can be a constant, a variable, or a combination of both. When multiplying binomials, each term from one binomial interacts with each term in the other, requiring careful application of the distributive property or methods like FOIL.
Understanding binomial expressions is crucial for simplifying polynomial expressions and solving algebraic equations efficiently.

Combining Like Terms

Combining like terms is an essential simplification process in algebra. Like terms are terms in a polynomial that have the same variables raised to the same powers. In the expression \(20x^2 - 14x - 10x + 7\), the terms \(-14x\) and \(-10x\) are like terms because they both include the variable \(x\) with an exponent of 1.
To simplify, you combine these like terms by adding or subtracting their coefficients. In this case:

• \(-14x - 10x = -24x\)

This results in the simplified expression \(20x^2 - 24x + 7\). Combining like terms helps to shorten and clarify polynomial expressions, making them easier to interpret and solve.