Question

1. do you notice a pattern when you multiply each of the following:

a. $(x + 1)(x - 1)$
b. $(x + 2)(x - 2)$
c. $(x + 3)(x - 3)$
d. $(4x + 5)(4x - 5)$

1. find the perimeter and area of the rectangle.

(rectangle with sides $3x + 5$ and $x - 1$)

Explanation:

Problem 15 (a - d)

Step 1: Multiply (x + 1)(x - 1)

Using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), here \(a = x\), \(b = 1\). So \((x + 1)(x - 1)=x^2-1^2=x^2 - 1\).

Step 2: Multiply (x + 2)(x - 2)

Using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), here \(a = x\), \(b = 2\). So \((x + 2)(x - 2)=x^2-2^2=x^2 - 4\).

Step 3: Multiply (x + 3)(x - 3)

Using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), here \(a = x\), \(b = 3\). So \((x + 3)(x - 3)=x^2-3^2=x^2 - 9\).

Step 4: Multiply (4x + 5)(4x - 5)

Using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), here \(a = 4x\), \(b = 5\). So \((4x + 5)(4x - 5)=(4x)^2-5^2 = 16x^2-25\).

Pattern Notice:

All these products follow the difference of squares pattern \((a + b)(a - b)=a^2 - b^2\), where in parts (a)-(c), \(a=x\) and \(b\) is 1, 2, 3 respectively, and in part (d), \(a = 4x\) and \(b = 5\).

Problem 16: Perimeter and Area of Rectangle

Step 1: Recall Formulas

• Perimeter of a rectangle: \(P = 2\times(\text{length}+\text{width})\)
• Area of a rectangle: \(A=\text{length}\times\text{width}\)

Given length \(l=3x + 5\) and width \(w=x - 1\).

Step 2: Calculate Perimeter

Substitute \(l\) and \(w\) into the perimeter formula:

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Step 3: Calculate Area

Substitute \(l\) and \(w\) into the area formula:

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Answer:

s:

• Problem 15:
• (a) \(x^2 - 1\)
• (b) \(x^2 - 4\)
• (c) \(x^2 - 9\)
• (d) \(16x^2 - 25\)
• Pattern: All use \((a + b)(a - b)=a^2 - b^2\) (difference of squares).
• Problem 16:
• Perimeter: \(8x + 8\)
• Area: \(3x^2+2x - 5\)