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):
\[

$$\begin{align*} (7x + 3)(x - 9) &= 7x \cdot x + 7x \cdot (-9) + 3 \cdot x + 3 \cdot (-9) \\ &= 7x^2 - 63x + 3x - 27 \\ &= 7x^2 - 60x - 27 \end{align*}$$

\]

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

Using the distributive property (FOIL method):
\[

$$\begin{align*} (5x + 3)(x + 1) &= 5x \cdot x + 5x \cdot 1 + 3 \cdot x + 3 \cdot 1 \\ &= 5x^2 + 5x + 3x + 3 \\ &= 5x^2 + 8x + 3 \end{align*}$$

\]

Step 3: Find the difference (Top - Bottom)

\[

$$\begin{align*} (7x^2 - 60x - 27) - (5x^2 + 8x + 3) &= 7x^2 - 60x - 27 - 5x^2 - 8x - 3 \\ &= (7x^2 - 5x^2) + (-60x - 8x) + (-27 - 3) \\ &= 2x^2 - 68x - 30 \end{align*}$$

\]

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)

\[

$$\begin{align*} (4x^2 - 1) - (x^2 - 16) &= 4x^2 - 1 - x^2 + 16 \\ &= (4x^2 - x^2) + (-1 + 16) \\ &= 3x^2 + 15 \end{align*}$$

\]

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:
\[

$$\begin{align*} (4x + 1)(3x - 7) &= 4x \cdot 3x + 4x \cdot (-7) + 1 \cdot 3x + 1 \cdot (-7) \\ &= 12x^2 - 28x + 3x - 7 \\ &= 12x^2 - 25x - 7 \end{align*}$$

\]

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

Using the distributive property:
\[

$$\begin{align*} (3x - 4)(2x - 8) &= 3x \cdot 2x + 3x \cdot (-8) - 4 \cdot 2x + (-4) \cdot (-8) \\ &= 6x^2 - 24x - 8x + 32 \\ &= 6x^2 - 32x + 32 \end{align*}$$

\]

Step 3: Find the difference (Top - Bottom)

\[

$$\begin{align*} (12x^2 - 25x - 7) - (6x^2 - 32x + 32) &= 12x^2 - 25x - 7 - 6x^2 + 32x - 32 \\ &= (12x^2 - 6x^2) + (-25x + 32x) + (-7 - 32) \\ &= 6x^2 + 7x - 39 \end{align*}$$

\]

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)

\[

$$\begin{align*} (9x^2 - 6x + 1) - (x^2 + 12x + 36) &= 9x^2 - 6x + 1 - x^2 - 12x - 36 \\ &= (9x^2 - x^2) + (-6x - 12x) + (1 - 36) \\ &= 8x^2 - 18x - 35 \end{align*}$$

\]

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):
\[

$$\begin{align*} (x - 2)(x^2 + 8x - 1) &= x \cdot x^2 + x \cdot 8x + x \cdot (-1) - 2 \cdot x^2 - 2 \cdot 8x - 2 \cdot (-1) \\ &= x^3 + 8x^2 - x - 2x^2 - 16x + 2 \\ &= x^3 + (8x^2 - 2x^2) + (-x - 16x) + 2 \\ &= x^3 + 6x^2 - 17x + 2 \end{align*}$$

\]

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):
\[

$$\begin{align*} (x + 4)(x^2 - 5x + 9) &= x \cdot x^2 + x \cdot (-5x) + x \cdot 9 + 4 \cdot x^2 + 4 \cdot (-5x) + 4 \cdot 9 \\ &= x^3 - 5x^2 + 9x + 4x^2 - 20x + 36 \\ &= x^3 + (-5x^2 + 4x^2) + (9x - 20x) + 36 \\ &= x^3 - x^2 - 11x + 36 \end{align*}$$

\]

Step 3: Find the difference (Top - Bottom)

\[

$$\begin{align*} &(x^3 + 6x^2 - 17x + 2) - (x^3 - x^2 - 11x + 36) \\ &= x^3 + 6x^2 - 17x + 2 - x^3 + x^2 + 11x - 36 \\ &= (x^3 - x^3) + (6x^2 + x^2) + (-17x + 11x) + (2 - 36) \\ &= 7x^2 - 6x - 34 \end{align*}$$

\]

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:
\[

$$\begin{align*} &3x \cdot x^2 + 3x \cdot (-10x) + 3x \cdot 2 + 5 \cdot x^2 + 5 \cdot (-10x) + 5 \cdot 2 \\ &= 3x^3 - 30x^2 + 6x + 5x^2 - 50x + 10 \\ &= 3x^3 + (-30x^2 + 5x^2) + (6x - 50x) + 10 \\ &= 3x^3 - 25x^2 - 44x + 10 \end{align*}$$

\]

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

Using the distributive property:
\[

$$\begin{align*} &2x \cdot x^2 + 2x \cdot (-3x) + 2x \cdot (-7) - 1 \cdot x^2 - 1 \cdot (-3x) - 1 \cdot (-7) \\ &= 2x^3 - 6x^2 - 14x - x^2 + 3x + 7 \\ &= 2x^3 + (-6x^2 - x^2) + (-14x + 3x) + 7 \\ &= 2x^3 - 7x^2 - 11x + 7 \end{align*}$$

\]

Step 3: Find the difference (Top - Bottom)

\[

$$\begin{align*} &(3x^3 - 25x^2 - 44x + 10) - (2x^3 - 7x^2 - 11x + 7) \\ &= 3x^3 - 25x^2 - 44x + 10 - 2x^3 + 7x^2 + 11x - 7 \\ &= (3x^3 - 2x^3) + (-25x^2 + 7x^2) + (-44x + 11x) + (10 - 7) \\ &= x^3 - 18x^2 - 33x + 3 \end{align*}$$

\]

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: \(x^3 - 18x^2 - 33x + 3\)