Problem 32 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (4 x-3 y)(2 x+5 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(8x^2 + 14xy - 15y^2\).

Step 1: Apply the distributive property

To perform the operation \((4x - 3y)(2x + 5y)\), first, use the distributive property, also known as the FOIL (First, Outer, Inner, Last) method. We'll multiply each term in the first parenthesis by each term in the second parenthesis.

Step 2: Multiply the 'First' terms

Multiply the first terms from each parenthesis: \(4x \times 2x = 8x^2\).This results in the term \(8x^2\).

Step 3: Multiply the 'Outer' terms

Multiply the outer terms: \(4x \times 5y = 20xy\).This results in the term \(20xy\).

Step 4: Multiply the 'Inner' terms

Multiply the inner terms: \(-3y \times 2x = -6xy\).This results in the term \(-6xy\).

Step 5: Multiply the 'Last' terms

Multiply the last terms: \(-3y \times 5y = -15y^2\).This results in the term \(-15y^2\).

Step 6: Combine the like terms

Now, combine all the terms obtained: \(8x^2 + 20xy - 6xy - 15y^2\).Simplify by combining the like terms \(20xy\) and \(-6xy\) to get \(8x^2 + 14xy - 15y^2\).

Key Concepts

Distributive Property

The distributive property is fundamental when working with polynomials, and it allows you to break down expressions into simpler parts that can be easily managed. When you see an expression like

• a(b + c)

it means you should distribute the single term 'a' across the terms inside the parenthesis. In mathematical terms, this becomes:

• a imes b + a imes c.

This helps in expanding expressions and is particularly useful in polynomial operations like

• (4x - 3y)(2x + 5y).

This property allows every term in the first bracket to be multiplied by every term in the second bracket, ensuring that nothing is forgotten and everything is calculated efficiently. This systematic approach ensures that each part of the expression is accounted for in the final result.

FOIL Method

The FOIL method is a specific application of the distributive property tailored for multiplying two binomials. FOIL stands for

• First,
• Outer,
• Inner,
• Last.

These refer to the order of multiplication:- **First**: Multiply the first terms in each binomial. - **Outer**: Multiply the outermost terms.- **Inner**: Multiply the innermost terms. - **Last**: Multiply the last terms in each binomial.
For the polynomial

• (4x - 3y)(2x + 5y),

applying FOIL:

• First: \(4x \times 2x = 8x^2\)
• Outer: \(4x \times 5y = 20xy\)
• Inner: \(-3y \times 2x = -6xy\)
• Last: \(-3y \times 5y = -15y^2\)

The FOIL method ensures each combination of terms is considered, simplifying the task of multiplying binomials and paving the way for further simplification.

Simplifying Expressions

Once we have expanded expressions using the distributive property or the FOIL method, our goal is to simplify them. Simplification involves writing the expression in its simplest form. For example, if we multiply

• (4x - 3y)(2x + 5y)

and have the expanded expression

• 8x^2 + 20xy - 6xy - 15y^2,

we need to combine the terms to simplify it. Simplification might involve combining like terms, factoring, or making minor adjustments in notation. This process makes it easier to interpret and work with the expression, as it reduces complexity and potential errors in calculation.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions. Like terms are terms in an expression that have the same variables raised to the same power. This means we can add or subtract their coefficients. For instance, in the expression:

• 8x^2 + 20xy - 6xy - 15y^2,

the terms 20xy and -6xy are like terms because they both have the variable 'xy'. Combining like terms involves:

• Aligning all terms that have identical variables and powers.
• Add or subtract their coefficients accordingly.
• Rewriting the equation in its simplified form.

When we combine 20xy and -6xy, we perform the subtraction of their coefficients which results in 14xy, thereby simplifying the expression to

• 8x^2 + 14xy - 15y^2.

This not only shortens the expression but also makes further operations more straightforward. It's a vital skill in making algebraic expressions manageable and comprehensible.