contestada

Complete the following exercises by applying polynomial identities to complex numbers. Show your work:
1. Factor x2 + 64. Check your work
2. Factor 16x2 + 49. Check your work
3. Find the product of (x + 9i))
4. Find the product of (x − 2i)2
5. Find the product of (x + (3+5i))2

Respuesta :

clothesmakeridk11

1.)

  =(x-8i)(x+8i)

x^2+8ix-8ix-64i^2

x^2-64i^2

x^2-64(-1)

x^2+64

2.)

   =(4x-7i)(4x+7i)

16x^2+28ix-28ix-49i^2

16x^2-49i^2

16x^2-49(-1)

16x^2+49

3.)

   =(x+9i)(x+9i)

x^2+9ix+9ix+81i^2

x^2+18ix+81(-1)

x^2+18ix-81

4.)

   =(x-2i)(x-2i)

x^2-2ix-2ix+4i^2

x^2-4ix+4(-1)

x^2-4ix-4

5.)

   =[x+(3+5i)]^2

(x+5i+3)^2

(x+5i+3)(x+5i+3)

x^2+5ix+3x+5ix+25i^2+15i+3x+15i+9

x^2+6x+10ix+30i+25i^2+9

x^2+6x+10ix+30i+25(-1)+9

x^2+6x+10ix+30i-25+9

x^2+6x+10ix+30i-16

Hope this helps :)

MrRoyal

Polynomial identities are used to simplify equations

  • The factored expression is (x- 8i)(x + 8i)
  • The factored expression is (4x- 7i)(4x + 7i)
  • The product (x + 9i)^2 is x^2+18ix-81
  • The product (x - 2i)^2 is x^2-4ix-4
  • The product is (x+(3 + 5i))^2 is x^2 +6x + 10ix + 30i -16

(1) Factor x^2 + 64

We have:

[tex]\mathbf{x^2 + 64}[/tex]

Express 64 as 8^2

[tex]\mathbf{x^2 + 64 = x^2 + 8^2}[/tex]

Rewrite the expression as

[tex]\mathbf{x^2 + 64 = x^2 - (8i)^2}[/tex]

Apply difference of two squares

[tex]\mathbf{x^2 + 64 = (x- 8i)(x + 8i)}[/tex]

Hence, the factored expression is (x- 8i)(x + 8i)

(2) Factor 16x^2 + 49

We have:

[tex]\mathbf{16x^2 + 49}[/tex]

Express 16 as 4^2, and 49 as 7^2

[tex]\mathbf{16x^2 + 49 = (4x)^2 + 7^2}[/tex]

Rewrite the expression as

[tex]\mathbf{16x^2 + 49 = (4x)^2 - (7i)^2}[/tex]

Apply difference of two squares

[tex]\mathbf{16x^2 + 49 = (4x- 7i)(4x + 7i)}[/tex]

Hence, the factored expression is (4x- 7i)(4x + 7i)

(3) The product of (x + 9i)^2

We have:

[tex]\mathbf{(x + 9i)^2}[/tex]

Rewrite as:

[tex]\mathbf{(x + 9i)^2 = (x+9i)(x+9i)}[/tex]

Expand

[tex]\mathbf{(x + 9i)^2 = x^2+9ix+9ix+81i^2}[/tex]

Evaluate like terms

[tex]\mathbf{(x + 9i)^2 = x^2+18ix+81i^2}[/tex]

Express i^2 as -1

[tex]\mathbf{(x + 9i)^2 = x^2+18ix+81(-1)}[/tex]

Evaluate the product

[tex]\mathbf{(x + 9i)^2 = x^2+18ix-81}[/tex]

Hence, the product (x + 9i)^2 is x^2+18ix-81

(4) The product of (x - 2i)^2

We have:

[tex]\mathbf{(x - 2i)^2}[/tex]

Rewrite as:

[tex]\mathbf{(x - 2i)^2 = (x - 2i)(x - 2i)}[/tex]

Expand

[tex]\mathbf{(x - 2i)^2 = x^2-2ix-2ix+4i^2}[/tex]

Evaluate like terms

[tex]\mathbf{(x - 2i)^2 = x^2-4ix+4i^2}[/tex]

Express i^2 as -1

[tex]\mathbf{(x - 2i)^2 = x^2-4ix+4(-1)}[/tex]

Evaluate the product

[tex]\mathbf{(x - 2i)^2 = x^2-4ix-4}[/tex]

Hence, the product (x - 2i)^2 is x^2-4ix-4

 (5) The product of  (x+(3+5i))^2

We have

[tex]\mathbf{(x+(3 + 5i))^2}[/tex]

Rewrite as:

[tex]\mathbf{(x+(3 + 5i))^2 = (x+3 + 5i)(x+ 3 + 5i)}[/tex]

Expand

[tex]\mathbf{(x+(3 + 5i))^2 = x(x+3 + 5i) +3(x+ 3 + 5i) +5i(x+ 3 + 5i)}[/tex]

[tex]\mathbf{(x+(3 + 5i))^2 = x^2 +3x + 5ix +3x+ 9 + 15i +5ix+ 15i + 25i^2}[/tex]

Collect like terms

[tex]\mathbf{(x+(3 + 5i))^2 = x^2 +3x +3x + 5ix +5ix + 15i + 15i + 25i^2+ 9}[/tex]

Express i^2 as -1

[tex]\mathbf{(x+(3 + 5i))^2 = x^2 +3x +3x + 5ix +5ix + 15i + 15i + 25(-1)+ 9}[/tex]

[tex]\mathbf{(x+(3 + 5i))^2 = x^2 +3x +3x + 5ix +5ix + 15i + 15i - 25+ 9}[/tex]

[tex]\mathbf{(x+(3 + 5i))^2 = x^2 +6x + 10ix + 30i -16}[/tex]

Hence, the product is (x+(3 + 5i))^2 is x^2 +6x + 10ix + 30i -16

Read more about polynomial identities at:

https://brainly.com/question/9514416

Otras preguntas