2 Polynomials; Question 2.19

[Eigenvalue]

2.19. Perform the indicated operations:

a) (x-h)²+(y-k)²
b) (x+h)⁴-x⁴
c) R²-(R-x)²
d) (ax+by+c)²

[Eigenvalue]

a) (x-h)²+(y-k)²
Use FOIl to expand:
x²-2xh+h²+y²-2yk+k²

[Eigenvalue]

b) (x+h)⁴-x⁴
Use binomial expansion:
x⁴+4x³h+6x²h²+4xh³+y⁴-x⁴
=4x³h+6x²h²+4xh³+y⁴

Note: the coefficients for binomial expansion can be found via Pascal's triangle:
(x+y)²=1x²+2xy+1y²=x²+2xy+y²

(x+y)³=1x³+3x²y+3xy²+1y³
=x³+3x²y+3xy²+y³

(x+y)⁴=1x⁴+4x³y+6x²y²+4xy³+1y⁴
=x⁴+4x³y+6x²y²+4xy³+y⁴

(x+y)⁵=1x⁵+5x⁴y+10x³y²+10x²y³+5xy⁴+1y⁵
=x⁵+5x⁴y+10x³y²+10x²y³+5xy⁴+y⁵

[Eigenvalue]

c) R²-(R-x)²
Use FOIL to expand:
R²-(R²-2Rx+x²)
=2Rx-x²

d) (ax+by+c)²
Use FOIL to expand:
(ax+by+c)(ax+by+c)²
=a²x²+abxy+acx+abxy+b²y²+cby+acx+cby+c²
=a²x²+2abxy+b²y²+2acx+2cby+c²