Question

set 2 simplify each expression. then, find the difference of your answers in each row.
1 (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 one of the problems, say problem 1: \((7x + 3)(x - 9)\) and \((5x + 3)(x + 1)\), then find their difference.

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

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 \((5x + 3)(x + 1)\)

Using the distributive property (FOIL method):
\[

$$\begin{align*} (5x + 3)(x + 1)&=5x\cdot x+5x\cdot1+3\cdot x+3\cdot1\\ &=5x^{2}+5x + 3x + 3\\ &=5x^{2}+8x + 3 \end{align*}$$

\]

Step 3: Find the difference \((7x^{2}-60x - 27)-(5x^{2}+8x + 3)\)

Subtract the second expanded form from the first:
\[

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

\]

Let's solve problem 8: \((2x + 1)(2x - 1)\) and \((x + 4)(x - 4)\), then find their difference.

Step 1: Expand \((2x + 1)(2x - 1)\)

Using the difference of squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 2x\) and \(b = 1\):
\[
(2x + 1)(2x - 1)=(2x)^{2}-1^{2}=4x^{2}-1
\]

Step 2: Expand \((x + 4)(x - 4)\)

Using the difference of squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a=x\) and \(b = 4\):
\[
(x + 4)(x - 4)=x^{2}-4^{2}=x^{2}-16
\]

Step 3: Find the difference \((4x^{2}-1)-(x^{2}-16)\)

Subtract the second expanded form from the first:
\[

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

\]

Let's solve problem 9: \((4x + 1)(3x - 7)\) and \((3x - 4)(2x - 8)\), then find their difference.

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

Using the distributive property:
\[

$$\begin{align*} (4x + 1)(3x - 7)&=4x\cdot3x+4x\cdot(-7)+1\cdot3x + 1\cdot(-7)\\ &=12x^{2}-28x+3x - 7\\ &=12x^{2}-25x - 7 \end{align*}$$

\]

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

Using the distributive property:
\[

$$\begin{align*} (3x - 4)(2x - 8)&=3x\cdot2x+3x\cdot(-8)-4\cdot2x-4\cdot(-8)\\ &=6x^{2}-24x-8x + 32\\ &=6x^{2}-32x + 32 \end{align*}$$

\]

Step 3: Find the difference \((12x^{2}-25x - 7)-(6x^{2}-32x + 32)\)

Subtract the second expanded form from the first:
\[

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

\]

Let's solve problem 10: \((3x - 1)^{2}\) and \((x + 6)^{2}\), then find their difference.

Step 1: Expand \((3x - 1)^{2}\)

Using the perfect square formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = 3x\) and \(b = 1\):
\[
(3x - 1)^{2}=(3x)^{2}-2\cdot3x\cdot1+1^{2}=9x^{2}-6x + 1
\]

Step 2: Expand \((x + 6)^{2}\)

Using the perfect square formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), where \(a=x\) and \(b = 6\):
\[
(x + 6)^{2}=x^{2}+2\cdot x\cdot6+6^{2}=x^{2}+12x + 36
\]

Step 3: Find the difference \((9x^{2}-6x + 1)-(x^{2}+12x + 36)\)

Subtract the second expanded form from the first:
\[

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

\]

Let's solve problem 11: \((x - 2)(x^{2}+8x - 1)\) and \((x + 4)(x^{2}-5x + 9)\), then find their difference.

Step 1: Expand \((x - 2)(x^{2}+8x - 1)\)

Using the distributive property:
\[

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

\]

Step 2: Expand \((x + 4)(x^{2}-5x + 9)\)

Using the distributive property:
\[

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

\]

Step 3: Find the difference \((x^{3}+6x^{2}-17x + 2)-(x^{3}-x^{2}-11x + 36)\)

Subtract the second expanded form from the first:
\[

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

\]

Let's solve problem 12: \((3x + 5)(x^{2}-10x + 2)\) and \((2x - 1)(x^{2}-3x - 7)\), then find their difference.

Step 1: Expand \((3x + 5)(x^{2}-10x + 2)\)

Using the distributive property:
\[

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

\]

Step 2: Expand \((2x - 1)(x^{2}-3x - 7)\)

Using the distributive property:
\[

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

\]

Step 3: Find the difference \((3x^{3}-25x^{2}-44x + 10)-(2x^{3}-7x^{2}-11x + 7)\)

Subtract the second expanded form from the first:
\[

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

\]