💡 12th Grade Other: Multiplication Practice Questions

This is a standard binomial multiplication problem. We can use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last).

• First: Multiply the first terms of each binomial: \( (2x)(x) = 2x^2 \).
• Outer: Multiply the outer terms: \( (2x)(-5) = -10x \).
• Inner: Multiply the inner terms: \( (3)(x) = 3x \).
• Last: Multiply the last terms: \( (3)(-5) = -15 \).

Now, combine these results and simplify by adding like terms:

💡 Key Concept: The distributive property is fundamental for multiplying polynomials.

To multiply a trinomial by a binomial, we distribute each term of the binomial to each term of the trinomial.

• Multiply \( x \) by each term in \( (x^2 - 2x + 1) \):
• \( x \cdot x^2 = x^3 \)
• \( x \cdot (-2x) = -2x^2 \)
• \( x \cdot 1 = x \)
• Multiply \( 4 \) by each term in \( (x^2 - 2x + 1) \):
• \( 4 \cdot x^2 = 4x^2 \)
• \( 4 \cdot (-2x) = -8x \)
• \( 4 \cdot 1 = 4 \)

Now, combine all the terms and group like terms:

Combine the \( x^2 \) terms and the \( x \) terms:

✅ Result: \( x^3 + 2x^2 - 7x + 4 \)

When multiplying terms with exponents, we multiply the coefficients and add the exponents of the same variables.

• Multiply the coefficients: \( 3 \times 5 = 15 \).
• Multiply the \( a \) terms: \( a^2 \times a^4 = a^{2+4} = a^6 \).
• Multiply the \( b \) terms: \( b^3 \times b^1 = b^{3+1} = b^4 \).

Combine these parts:

📌 Rule: \( x^m \cdot x^n = x^{m+n} \)

First, let's expand \( (x+y)^2 \):

Using the distributive property (FOIL):

Since \( xy = yx \), we combine like terms:

Now, let's look at \( (x+y)(x+y+z) \). We can distribute \( (x+y) \) to each term in the second parenthesis:

Expanding each part:

Combining like terms:

Connection: Notice that the first three terms \( x^2 + 2xy + y^2 \) are exactly the expansion of \( (x+y)^2 \). This shows that when multiplying \( (x+y) \) by \( (x+y+z) \), we are essentially taking the expansion of \( (x+y)^2 \) and then adding the terms that result from multiplying \( (x+y) \) by \( z \).

💡 Insight: Understanding the multiplication of simpler expressions can provide a framework for more complex ones.

The area of a rectangle is calculated by multiplying its length by its width.

• Length: \( L = (2w + 3) \) feet
• Width: \( W = (w - 1) \) feet
• Area: \( A = L \times W \)

Substitute the expressions for length and width:

Now, use the distributive property (FOIL) to multiply these binomials:

• First: \( (2w)(w) = 2w^2 \)
• Outer: \( (2w)(-1) = -2w \)
• Inner: \( (3)(w) = 3w \)
• Last: \( (3)(-1) = -3 \)

Combine and simplify:

The expression for the total area of the garden bed is \( (2w^2 + w - 3) \) square feet.

👉 Application: This is a common application in landscaping and construction where dimensions are often expressed algebraically.

To multiply three binomials, it's easiest to multiply two at a time, and then multiply the result by the third.

• Step 1: Multiply the first two binomials \( (x - 1)(x + 2) \).
• Using FOIL: \( x \cdot x + x \cdot 2 + (-1) \cdot x + (-1) \cdot 2 \)
• \( x^2 + 2x - x - 2 \)
• \( x^2 + x - 2 \)
• Step 2: Now multiply the result \( (x^2 + x - 2) \) by the third binomial \( (x + 3) \).
• Distribute each term of \( (x + 3) \) to \( (x^2 + x - 2) \).
• \( x(x^2 + x - 2) + 3(x^2 + x - 2) \)
• \( (x^3 + x^2 - 2x) + (3x^2 + 3x - 6) \)

Combine like terms:

✅ Final Answer: \( x^3 + 4x^2 + x - 6 \)

Pattern Observation: The pattern suggests that multiplying a binomial of the form \( (x-a) \) by \( (x+a) \) results in \( x^2 - a^2 \).

Algebraic Explanation:

We can use the distributive property (FOIL) to multiply \( (x-a)(x+a) \):

• First: \( x \cdot x = x^2 \)
• Outer: \( x \cdot a = ax \)
• Inner: \( (-a) \cdot x = -ax \)
• Last: \( (-a) \cdot a = -a^2 \)

Combine the terms:

The middle terms, \( ax \) and \( -ax \), cancel each other out:

💡 Key Identity: This is known as the difference of squares identity. It's a crucial pattern in algebra for simplifying expressions and factoring polynomials.

The revenue \( R \) is defined as the product of the number of units sold and the price per unit.

• Number of Units Sold: \( (100 - 2p) \)
• Price Per Unit: \( p \)
• Revenue: \( R = (\text{Number of Units Sold}) \times (\text{Price Per Unit}) \)

Substitute the given expressions:

Now, distribute \( p \) to each term inside the parenthesis:

The expression for the monthly revenue is \( R = 100p - 2p^2 \) dollars.

📌 Context: This type of expression is fundamental in economics and business for understanding how pricing affects revenue and profit.