Giải bài 5 trang 28 SGK Toán 8 tập 1 - Cánh diều

Đề bài

Phân tích mỗi đa thức sau thành nhân tử:

a) ${\left( {x + 2y} \right)^2} - {\left( {x - y} \right)^2}$                                           

b) ${\left( {x + 1} \right)^3} + {\left( {x - 1} \right)^3}$

c) $9{x^2} - 3x + 2y - 4{y^2}$                                   

d) $4{x^2} - 4xy + 2x - y + {y^2}$

e) ${x^3} + 3{{\rm{x}}^2} + 3{\rm{x}} + 1 - {y^3}$                                        

g) ${x^3} - 2{{\rm{x}}^2}y + x{y^2} - 4{\rm{x}}$

Lời giải chi tiết

a)

$\begin{array}{l}{\left( {x + 2y} \right)^2} - {\left( {x - y} \right)^2}\\ = \left( {x + 2y + x - y} \right)\left( {x + 2y - x + y} \right)\\ = \left( {2{\rm{x}} + y} \right).3y\end{array}$     

b) 

$\begin{array}{l}{\left( {x + 1} \right)^3} + {\left( {x - 1} \right)^3}\\ = \left( {x + 1 + x - 1} \right)\left[ {{{\left( {x + 1} \right)}^2} - \left( {x + 1} \right)\left( {x - 1} \right) + {{\left( {x - 1} \right)}^2}} \right]\\ = 2{\rm{x}}\left[ {{x^2} + 2{\rm{x}} + 1 - \left( {{x^2} - 1} \right) + {x^2} - 2{\rm{x}} + 1} \right]\\ = 2{\rm{x}}\left( {{x^2} + 2{\rm{x}} + 1 - {x^2} + 1 + {x^2} - 2{\rm{x}} + 1} \right)\\ = 2{\rm{x}}\left( {{x^2} + 3} \right)\end{array}$        

c)  

$\begin{array}{l}9{x^2} - 3x + 2y - 4{y^2}\\ = \left( {9{x^2} - 4{y^2}} \right) - \left( {3x - 2y} \right)\\ = \left( {3x - 2y} \right)\left( {3x + 2y} \right) - \left( {3x - 2y} \right)\\ = \left( {3x - 2y} \right)\left( {3x + 2y - 1} \right)\end{array}$

d)     

$\begin{array}{l}4{x^2} - 4xy + 2x - y + {y^2}\\ = \left( {4{x^2} - 4xy + {y^2}} \right) + \left( {2x - y} \right)\\ = {\left( {2x - y} \right)^2} + \left( {2x - y} \right)\\ = \left( {2x - y} \right)\left( {2x - y + 1} \right)\end{array}$

e)

$\begin{array}{l}{x^3} + 3{{\rm{x}}^2} + 3{\rm{x}} + 1 - {y^3}\\ = \left( {{x^3} + 3{{\rm{x}}^2} + 3{\rm{x}} + 1} \right) - {y^3}\\ = {\left( {x + 1} \right)^3} - {y^3}\\ = \left( {x + 1 - y} \right)\left[ {{{\left( {x + 1} \right)}^2} + \left( {x + 1} \right)y + {y^2}} \right]\end{array}$

g)

$\begin{array}{l}{x^3} - 2{{\rm{x}}^2}y + x{y^2} - 4{\rm{x}}\\{\rm{ = }}\left( {{x^3} - 2{{\rm{x}}^2}y + x{y^2}} \right) - 4{\rm{x}}\\ = x\left( {{x^2} - 2{\rm{x}}y + {y^2}} \right) - 4{\rm{x}}\\ = x{\left( {x - y} \right)^2} - 4{\rm{x}}\\ = x\left[ {{{\left( {x - y} \right)}^2} - {2^2}} \right]\\ = x\left( {x - y + 2} \right)\left( {x - y - 2} \right)\end{array}$