rút gọn biểu thức : (x^2 -5x+1)^2 +(5x-1)^2 +(5x-1)(x^2 -5x+1) 13/07/2021 Bởi Elliana rút gọn biểu thức : (x^2 -5x+1)^2 +(5x-1)^2 +(5x-1)(x^2 -5x+1)
Đặt $\begin{array}{l} \left\{ \begin{array}{l} a = {x^2} – 5x + 1\\ b = 5x – 1 \to a + b = {x^2} \end{array} \right.\\ BT \to {a^2} + {b^2} + ab\\ = {a^2} + {b^2} + 2ab\\ = {\left( {a + b} \right)^2} – ab\\ = {\left( {{x^2}} \right)^2} – \left( {{x^2} – 5x + 1} \right)\left( {5x – 1} \right)\\ = {x^4} – \left( {5{x^3} – {x^2} – 25{x^2} + 5x + 5x – 1} \right)\\ = {x^4} – 5{x^3} + 26{x^2} – 10x + 1 \end{array}$ Bình luận
`(x^2 – 5x + 1)^2 + (5x – 1)^2 + (5x – 1)(x^2 – 5x + 1)` `= (x^2 – 5x + 1)^2 + 2(5x – 1)(x^2 – 5x + 1) + (5x – 1)^2 – (5x – 1)(x^2 – 5x + 1)`Áp dụng đẳng thức `x^2 + 2xy + y^2 = (x + y)^2`, biểu thức trở thành: `(x^2 – 5x + 1 + 5x – 1)^2 – (5x – 1)(x^2 – 5x + 1)` `= (x^2)^2 – (5x. x^2 – 5x. 5x + 5x – x^2 + 5x – 1)` `= x^4 – (5x^3 – 25x^2 + 5x – x^2 + 5x – 1)` `= x^4 – (5x^3 – 26x^2 + 10x – 1)` `= x^4 – 5x^3 + 26x^2 – 10x + 1` Bình luận
Đặt
$\begin{array}{l} \left\{ \begin{array}{l} a = {x^2} – 5x + 1\\ b = 5x – 1 \to a + b = {x^2} \end{array} \right.\\ BT \to {a^2} + {b^2} + ab\\ = {a^2} + {b^2} + 2ab\\ = {\left( {a + b} \right)^2} – ab\\ = {\left( {{x^2}} \right)^2} – \left( {{x^2} – 5x + 1} \right)\left( {5x – 1} \right)\\ = {x^4} – \left( {5{x^3} – {x^2} – 25{x^2} + 5x + 5x – 1} \right)\\ = {x^4} – 5{x^3} + 26{x^2} – 10x + 1 \end{array}$
`(x^2 – 5x + 1)^2 + (5x – 1)^2 + (5x – 1)(x^2 – 5x + 1)`
`= (x^2 – 5x + 1)^2 + 2(5x – 1)(x^2 – 5x + 1) + (5x – 1)^2 – (5x – 1)(x^2 – 5x + 1)`
Áp dụng đẳng thức `x^2 + 2xy + y^2 = (x + y)^2`, biểu thức trở thành:
`(x^2 – 5x + 1 + 5x – 1)^2 – (5x – 1)(x^2 – 5x + 1)`
`= (x^2)^2 – (5x. x^2 – 5x. 5x + 5x – x^2 + 5x – 1)`
`= x^4 – (5x^3 – 25x^2 + 5x – x^2 + 5x – 1)`
`= x^4 – (5x^3 – 26x^2 + 10x – 1)`
`= x^4 – 5x^3 + 26x^2 – 10x + 1`