Представьте многочлен в виде произведения. 1) m3 - n3 + 2n - 2m 3) x6 + y6 + x2 + y2 5)...

Question

Представьте многочлен в виде произведения. 1) m3 - n3 + 2n - 2m 3) x6 + y6 + x2 + y2 5) x4 + xy3 - x3y - y4

Answer 1

1) m³ - n³ + 2n - 2m = ( m³ - n³) + 2( n -m) = ( m - n)(m² +mn + n²) - 2( m - n) =
= ( m - n)(m² + mn + n² - 2)
2) x⁶ + y⁶ + x² + y² = (x²)³ + (y²)³ + ( x² + y²) = ( x² + y²)(x⁴ - x²y² + y⁴) + ( x² + y²) =
= (x² + y²)( x⁴ - x²y² + y⁴ + 1)
3) x⁴ + xy3 - x3y - y⁴ = x⁴ + 3xy - 3xy - y⁴ = x⁴ - y⁴ = ( x²)² - (y²)² = (x² - y²)(x² + y²).

Answer 2

1)  m³-n³+2n-2m=(m³-n³)-2(n-m)=(n-m)(n²+mn+m²)-2(n-m)=(n-m)(n²+nm+m²-2)
3) x^6+y^6+x²+y²=(x²)³+(y²)³+ (x²+y²)=((x²+y²)(x^4-x²y²+y^4)+(x²+y²)=(x²+y²)(x^4-
-x²y²+y^4+1)
5) x^4+xy³-x³y-y^4=(x^4-y^4)-(x³y-xy³)=(x²-y²)(x²+y²)-xy(x²-y²)=(x²-y²)(x²+y²-xy)