Question

set 2: simplify each expression. then, find the difference of your answers in each row.
7 ((7x + 3)(x - 9)) ((5x + 3)(x + 1))
8 ((2x + 1)(2x - 1)) ((x + 4)(x - 4))
9 ((4x + 1)(3x - 7)) ((3x - 4)(2x - 8))
10 ((3x - 1)^2) ((x + 6)^2)
11 ((x - 2)(x^2 + 8x - 1)) ((x + 4)(x^2 - 5x + 9))
12 ((3x + 5)(x^2 - 10x + 2)) ((2x - 1)(x^2 - 3x - 7))

Answer:

Let's solve these problems one by one. We'll start with row 7 (problem 7) and then do the corresponding bottom expression, then find the difference.

Problem 7 (Top: \((7x + 3)(x - 9)\); Bottom: \((5x + 3)(x + 1)\))

Step 1: Expand the top expression \((7x + 3)(x - 9)\)

Using the distributive property (FOIL method):

$$
LATEXBLOCK0
$$

Step 2: Expand the bottom expression \((5x + 3)(x + 1)\)

Using the distributive property (FOIL method):

$$
LATEXBLOCK1
$$

Step 3: Find the difference (Top - Bottom)

$$
LATEXBLOCK2
$$

Problem 8 (Top: \((2x + 1)(2x - 1)\); Bottom: \((x + 4)(x - 4)\))

Step 1: Expand the top expression \((2x + 1)(2x - 1)\)

This is a difference of squares: \((a + b)(a - b) = a^2 - b^2\), where \(a = 2x\) and \(b = 1\):

$$ (2x + 1)(2x - 1) = (2x)^2 - 1^2 = 4x^2 - 1 $$

Step 2: Expand the bottom expression \((x + 4)(x - 4)\)

This is also a difference of squares: \((a + b)(a - b) = a^2 - b^2\), where \(a = x\) and \(b = 4\):

$$ (x + 4)(x - 4) = x^2 - 4^2 = x^2 - 16 $$

Step 3: Find the difference (Top - Bottom)

$$
LATEXBLOCK3
$$

Problem 9 (Top: \((4x + 1)(3x - 7)\); Bottom: \((3x - 4)(2x - 8)\))

Step 1: Expand the top expression \((4x + 1)(3x - 7)\)

Using the distributive property:

$$
LATEXBLOCK4
$$

Step 2: Expand the bottom expression \((3x - 4)(2x - 8)\)

Using the distributive property:

$$
LATEXBLOCK5
$$

Step 3: Find the difference (Top - Bottom)

$$
LATEXBLOCK6
$$

Problem 10 (Top: \((3x - 1)^2\); Bottom: \((x + 6)^2\))

Step 1: Expand the top expression \((3x - 1)^2\)

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), where \(a = 3x\) and \(b = 1\):

$$ (3x - 1)^2 = (3x)^2 - 2 \cdot 3x \cdot 1 + 1^2 = 9x^2 - 6x + 1 $$

Step 2: Expand the bottom expression \((x + 6)^2\)

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = x\) and \(b = 6\):

$$ (x + 6)^2 = x^2 + 2 \cdot x \cdot 6 + 6^2 = x^2 + 12x + 36 $$

Step 3: Find the difference (Top - Bottom)

$$
LATEXBLOCK7
$$

Problem 11 (Top: \((x - 2)(x^2 + 8x - 1)\); Bottom: \((x + 4)(x^2 - 5x + 9)\))

Step 1: Expand the top expression \((x - 2)(x^2 + 8x - 1)\)

Using the distributive property (multiply \(x\) and \(-2\) by each term in the quadratic):

$$
LATEXBLOCK8
$$

Step 2: Expand the bottom expression \((x + 4)(x^2 - 5x + 9)\)

Using the distributive property (multiply \(x\) and \(4\) by each term in the quadratic):

$$
LATEXBLOCK9
$$

Step 3: Find the difference (Top - Bottom)

$$
LATEXBLOCK10
$$

Problem 12 (Top: \((3x + 5)(x^2 - 10x + 2)\); Bottom: \((2x - 1)(x^2 - 3x - 7)\))

Step 1: Expand the top expression \((3x + 5)(x^2 - 10x + 2)\)

Using the distributive property:

$$
LATEXBLOCK11
$$

Step 2: Expand the bottom expression \((2x - 1)(x^2 - 3x - 7)\)

Using the distributive property:

$$
LATEXBLOCK12
$$

Step 3: Find the difference (Top - Bottom)

$$
LATEXBLOCK13
$$

Final Answers (Differences)

• Problem 7: \(2x^2 - 68x - 30\)
• Problem 8: \(3x^2 + 15\)
• Problem 9: \(6x^2 + 7x - 39\)
• Problem 10: \(8x^2 - 18x - 35\)
• Problem 11: \(7x^2 - 6x - 34\)
• Problem 12: \(x^3 - 18x^2 - 33x + 3\)