Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated operations and simplify. \((4 x-11)(3 x-7)\)

Step-by-Step Solution

Answer

The simplified result is \(12x^2 - 61x + 77\).

Step 1: Recognize the Operation

The given problem involves the multiplication of two binomials: \((4x-11)\) and \((3x-7)\). This requires the application of the distributive property or the FOIL method (First, Outside, Inside, Last) to expand the expression.

Step 2: Apply FOIL - First Terms

Multiply the first terms in each binomial: \(4x \times 3x = 12x^2\).

Step 3: Apply FOIL - Outside Terms

Multiply the outside terms: \(4x \times (-7) = -28x\).

Step 4: Apply FOIL - Inside Terms

Multiply the inside terms: \(-11 \times 3x = -33x\).

Step 5: Apply FOIL - Last Terms

Multiply the last terms: \(-11 \times (-7) = 77\).

Step 6: Combine Like Terms

Combine all the terms obtained from the FOIL steps: \(12x^2 - 28x - 33x + 77\). Then simplify by combining like terms for the linear x-term: \(12x^2 - 61x + 77\).

Step 7: Simplified Result

The expression has been fully simplified to: \(12x^2 - 61x + 77\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify the multiplication of expressions. It states that a single term outside of a parenthesis can be multiplied by each term inside the parenthesis separately. This property is often written as:

• If you have an expression like \(a(b + c)\), you distribute 'a' to both 'b' and 'c': \(ab + ac\).
• This can be extended to expressions with more terms, e.g., \(a(b + c + d)\) becomes \(ab + ac + ad\).

Applying this to polynomial multiplication, such as multiplying binomials, involves distributing each term in the first binomial across the terms of the second. For example, with the expression \((4x - 11)(3x - 7)\), you multiply each term in the first binomial by each term in the second binomial.
You would perform the following steps:

• 4x multiplied by 3x and then 4x multiplied by -7.
• Next, -11 is multiplied by both 3x and -7.

This strategic "breakdown" ensures each part of a binomial affects every part of another. This method lays the groundwork for more structured approaches, like the FOIL method, but is broader and applies to polynomials of any length.

FOIL Method

The FOIL Method is a straightforward technique designed to make binomial multiplication easier and more organized. It is essentially an application of the distributive property but focuses on binomials specifically. FOIL stands for: First, Outside, Inside, Last. Each word refers to a pair of terms you multiply from each binomial.

• First refers to multiplying the first terms from each binomial: For \((4x - 11)\) and \((3x - 7)\), multiply \(4x\) with \(3x\) to get \(12x^2\).
• Outside deals with the outer terms of the binomials: Multiply \(4x\) with \(-7\), resulting in \(-28x\).
• Inside addresses the inner terms: Multiply \(-11\) by \(3x\) to get \(-33x\).
• Last focuses on the last terms in each binomial: Multiply \(-11\) with \(-7\) to obtain \(77\).

Once all these interactions are completed, the resulting products are combined to yield the final expression.

In practice, combining these results gives:

• 12x^2 - 28x - 33x + 77.
• Simplify by combining like terms, especially those that share the 'x' variable, ending up with: \(12x^2 - 61x + 77\).

Utilizing the FOIL method not only makes complex binomial multiplication simpler but also reinforces an organizational strategy in algebraic manipulation.

Binomial Expansion

Binomial expansion is a concept that allows us to expand expressions raised to a power in the form \((a + b)^n\). It draws directly from principles such as the distributive property and organized methods like FOIL when dealing with second-degree polynomials (binomials).

When engaging in binomial expansion, especially with powers higher than 1, you'll use not just basic multiplication but tools like Pascal's Triangle or the Binomial Theorem, which greatly simplify the expansion process in higher powers.
Here’s a simplified approach:

• The binomial \((a + b)^2\) can be expanded using FOIL: First (\(a^2\)), Outside (\(ab\)), Inside (\(ba\)), and Last (\(b^2\)).
• When raised to higher powers, say \((x + y)^3\), the expansion would include terms like \(x^3 + 3x^2y + 3xy^2 + y^3\).
• Each term's coefficient in the expansion follows a pattern represented by the Binomial Coefficients.

Understanding binomial expansion is essential for tackling polynomial functions of higher degrees, simplifying complex expressions, and preparing for more advanced topics in algebra.