Question

1. use generic rectangles to multiply each of the following expressions. homework help

a. ((x + 2)(x - 5))
b. ((y + 2x)(y + 3x))
c. ((3y - 8)(-x + y))
d. ((x - 3y)(x + 3y))

Explanation:

Part a: \((x + 2)(x - 5)\)

Step1: Draw generic rectangle

Divide the rectangle into four parts with length \(x - 5\) and width \(x + 2\). The four sub - rectangles have dimensions: \(x\times x\), \(x\times(- 5)\), \(2\times x\), \(2\times(-5)\)

Step2: Calculate each sub - area

• Area of \(x\times x\): \(x\times x=x^{2}\)
• Area of \(x\times(-5)\): \(x\times(-5)=-5x\)
• Area of \(2\times x\): \(2\times x = 2x\)
• Area of \(2\times(-5)\): \(2\times(-5)=-10\)

Step3: Sum the areas

\(x^{2}-5x + 2x-10=x^{2}-3x - 10\)

Part b: \((y + 2x)(y + 3x)\)

Step1: Draw generic rectangle

Divide the rectangle into four parts with length \(y + 3x\) and width \(y + 2x\). The four sub - rectangles have dimensions: \(y\times y\), \(y\times3x\), \(2x\times y\), \(2x\times3x\)

Step2: Calculate each sub - area

• Area of \(y\times y\): \(y\times y=y^{2}\)
• Area of \(y\times3x\): \(y\times3x = 3xy\)
• Area of \(2x\times y\): \(2x\times y=2xy\)
• Area of \(2x\times3x\): \(2x\times3x = 6x^{2}\)

Step3: Sum the areas

\(y^{2}+3xy + 2xy+6x^{2}=y^{2}+5xy + 6x^{2}\)

Part c: \((3y - 8)(-x + y)\)

Step1: Draw generic rectangle

Divide the rectangle into four parts with length \(-x + y\) and width \(3y-8\). The four sub - rectangles have dimensions: \(3y\times(-x)\), \(3y\times y\), \(-8\times(-x)\), \(-8\times y\)

Step2: Calculate each sub - area

• Area of \(3y\times(-x)\): \(3y\times(-x)=-3xy\)
• Area of \(3y\times y\): \(3y\times y = 3y^{2}\)
• Area of \(-8\times(-x)\): \(-8\times(-x)=8x\)
• Area of \(-8\times y\): \(-8\times y=-8y\)

Step3: Sum the areas

\(-3xy + 3y^{2}+8x-8y=3y^{2}-3xy + 8x-8y\)

Part d: \((x - 3y)(x + 3y)\)

Answer:

s:
a. \(x^{2}-3x - 10\)

b. \(y^{2}+5xy + 6x^{2}\)

c. \(3y^{2}-3xy + 8x-8y\)

d. \(x^{2}-9y^{2}\)