Question

Chứng minh các đẳng thức

a) (x + a) . (x + b) = x²⁺ + (a + b) . x + ab

b) (x + a) . (x + b) . (x + c) = x³ + (a + b + c) . x² + (ab + bc + ca) . x + abc.

Answer 1

a ) VP = \(\left(x+a\right).\left(x+b\right)=x^2+bx+ax+ab\)

     VT = \(x^2+\left(a+b\right).x+ab=x^2+ax+bx+ab\)

\(\Rightarrow VT=VP\)

b ) VP : \(\left(x+a\right).\left(x+b\right)\left(x+c\right)=\left(x^2+bx+ax+ab\right).\left(x+c\right)\) ( Vế đầu áp dụng luôn ở câu a )

\(=x^2.x+x^2.c+bx.x+bx.c+ax.x+ax.c+ab.x+ab.c\)

\(=x^3+cx^2+bx^2+cbx+ax^2+cax+abx+abc\)

\(=x^3+\left(cx^2+bx^2+ax^2\right)+\left(cbx+cax+abx\right)+abc\)

\(=x^3+\left(a+b+c\right)x^2+\left(ab+ac+bc\right).x+abc\)

Vậy \(\left(x+a\right).\left(x+b\right).\left(x+c\right)=x^3+\left(a+b+c\right).x^2+\left(ab+ca+bc\right).x+abc\)

Answer 2

a) VP =\(\left(x+a\right)\left(x+b\right)=x^2+bx+\text{ax+ab}\)

\(VT=x^2+\left(a+b\right).x+ab=x^2+ax+bx+ab\\ =>VT=VP\)

b) VP : \(\left(x+a\right).\left(x+b\right).\left(x+c\right)=\left(x^2+bx+ax+ab\right).\left(x+c\right)\)( Vế đầu áp dụng luôn ở câu a )

\(=x^2.x+x^2.c+bx.x+bx.c+\text{ax}.x+\text{ax}.c+ab.c+ab.c\\ =x^3+cx^2+bx^2-cbx+\text{ax}^2+ca.x+ab.x+abc\\ \)

\(=x^3+\left(cx^2+bx^2+\text{ax}^2\right)-\left(cbx+c\text{ax}+abx\right)+abc\\ =x^3-\left(a+b+c\right)x^2+\left(ab+ac+bc\right).x+abc\)

Vậy \(\left(x+a\right)\left(x-b\right)\left(x+c\right)=x^3+\left(a+b+c\right).x^2+\left(ab+ca+bc\right).x+abc\)