Write the degree of the following polynomials.
(a) 5xy + 8x'y + 2xy +9yco (b) 16p'q +32pq' + Spa
(d) ry - xy +7xy - 39 (e) 6xy - 4x - y + 5
Answers
Given:
[math]x^2 + y^2 = 20[/math]
[math]xy = 8[/math]
Evaluate:
[math]x^2 - y^2[/math]
Use these algebraic identities:
[math]x^2 + 2xy+ y^2 =(x+y)^2[/math]
[math]x^2 - y^2 =(x+y)(x-y)[/math]
In the first of the two identities I listed, substitute known values for the variables on the left side.
[math]x^2 + 2xy+ y^2= (x+y)^2 [/math]
→ [math]20+2(8) = (x+y)^2 [/math]
→ [math]36 = (x+y)^2 [/math]
→[math]6^2 =(x+y)^2 [/math]
→[math]\pm6[/math][math]=(x+y) [/math]
From xy = 8, we know y= 8/x.
→[math] \pm 6 = x+ \frac{8}{x}[/math]
→[math] \pm 6 = \frac{x^2+8}{x}[/math]
→[math]\pm 6x = x^2+8[/math]
[math]0 = x^2\pm6x+8[/math]
0 = (x+2)(x+4)
→ x=-2 or x=-4
or
0 = (x-2)(x-4)
→ x=2 or x=4
So, (x,y) = (-2,-4) or (-4,-2) or (2,4) or (4,2).
[math]x^2 - y^2 =(x+y)(x-y) = \pm12[/math]
The answer is
[math]x^2 - y^2 = \pm12[/math]