Дефиниција, решени задаци.
Задаци Текст задатака објашњених у видео лекцији:
Пр.1) Израчунати следеће производе:
а) 2 x ⋅ ( 3 x 2 + 5 x − 8 ) 2x \cdot \left( {3{x^2} + 5x - 8} \right) 2 x ⋅ ( 3 x 2 + 5 x − 8 )
б) − 8 n 5 ⋅ ( n 4 + n 2 + 7 n ) - 8{n^5} \cdot \left( {{n^4} + {n^2} + 7n} \right) − 8 n 5 ⋅ ( n 4 + n 2 + 7 n )
в) ( − 1 2 x 2 + 2 3 x ) ⋅ 6 x \left( { - \frac{1}{2}{x^2} + \frac{2}{3}x} \right) \cdot 6x ( − 2 1 x 2 + 3 2 x ) ⋅ 6 x
г) 1 , 2 m 3 ⋅ ( 4 1 6 m 2 + 5 6 m − 2 11 12 ) 1,2{m^3} \cdot \left( {4\frac{1}{6}{m^2} + \frac{5}{6}m - 2\frac{{11}}{{12}}} \right) 1 , 2 m 3 ⋅ ( 4 6 1 m 2 + 6 5 m − 2 12 11 )
Пр.2) Упростити изразе:
а) ( − 3 x 3 ) 2 ⋅ ( x − 3 ) {\left( { - 3{x^3}} \right)^2} \cdot \left( {x - 3} \right) ( − 3 x 3 ) 2 ⋅ ( x − 3 )
б) ( 8 x 2 + 16 x − 4 ) ⋅ ( − 1 2 x 2 ) 3 \left( {8{x^2} + 16x - 4} \right) \cdot {\left( { - \frac{1}{2}{x^2}} \right)^3} ( 8 x 2 + 16 x − 4 ) ⋅ ( − 2 1 x 2 ) 3
в ( 0 , 1 x y ) 2 ⋅ ( 8 x 2 + 16 y − 4 y 2 ) {\left( {0,1xy} \right)^2} \cdot \left( {8{x^2} + 16y - 4{y^2}} \right) ( 0 , 1 x y ) 2 ⋅ ( 8 x 2 + 16 y − 4 y 2 ) Пр.3) Упростити изразе: а) x ⋅ ( 3 x 2 + 7 x − 10 ) + 3 x ⋅ ( 4 x 2 + 2 x − 1 ) x \cdot \left( {3{x^2} + 7x - 10} \right) + 3x \cdot \left( {4{x^2} + 2x - 1} \right) x ⋅ ( 3 x 2 + 7 x − 10 ) + 3 x ⋅ ( 4 x 2 + 2 x − 1 ) б) − 3 y 2 ⋅ ( 2 y 2 + 3 y + 4 ) − 6 y ⋅ ( y 3 − 2 y 2 + 7 y ) - 3{y^2} \cdot \left( {2{y^2} + 3y + 4} \right) - 6y \cdot \left( {{y^3} - 2{y^2} + 7y} \right) − 3 y 2 ⋅ ( 2 y 2 + 3 y + 4 ) − 6 y ⋅ ( y 3 − 2 y 2 + 7 y ) в) ( − 1 3 x 2 + 1 2 x ) ⋅ 6 x + 4 x 2 ⋅ ( − 1 2 x + 3 4 ) \left( { - \frac{1}{3}{x^2} + \frac{1}{2}x} \right) \cdot 6x + 4{x^2} \cdot \left( { - \frac{1}{2}x + \frac{3}{4}} \right) ( − 3 1 x 2 + 2 1 x ) ⋅ 6 x + 4 x 2 ⋅ ( − 2 1 x + 4 3 ) Пр.4) Квадрат монома 3 x 3x 3 x 2 x − 5 2x - 5 2 x − 5 добијени производ умањити за производ монома x 2 {x^2} x 2 бинома 4 x − 3 4x - 3 4 x − 3 Пр.1)
а) 2 x ⋅ ( 3 x 2 + 5 x − 8 ) = 2 x ⋅ 3 x 2 + 2 x ⋅ 5 x − 2 x ⋅ 8 = 6 x 3 + 10 x 2 − 16 x 2x \cdot \left( {3{x^2} + 5x - 8} \right) = 2x \cdot 3{x^2} + 2x \cdot 5x - 2x \cdot 8 = 6{x^3} + 10{x^2} - 16x 2 x ⋅ ( 3 x 2 + 5 x − 8 ) = 2 x ⋅ 3 x 2 + 2 x ⋅ 5 x − 2 x ⋅ 8 = 6 x 3 + 10 x 2 − 16 x
б) − 8 n 5 ⋅ ( n 4 + n 2 + 7 n ) = − 8 n 5 ⋅ n 4 − 8 n 5 ⋅ n 2 − 8 n 5 ⋅ 7 n = − 8 n 9 − 8 n 7 − 56 n 6 - 8{n^5} \cdot \left( {{n^4} + {n^2} + 7n} \right) = - 8{n^5} \cdot {n^4} - 8{n^5} \cdot {n^2} - 8{n^5} \cdot 7n = - 8{n^9} - 8{n^7} - 56{n^6} − 8 n 5 ⋅ ( n 4 + n 2 + 7 n ) = − 8 n 5 ⋅ n 4 − 8 n 5 ⋅ n 2 − 8 n 5 ⋅ 7 n = − 8 n 9 − 8 n 7 − 56 n 6
в) ( − 1 2 x 2 + 2 3 x ) ⋅ 6 x = − 1 2 x 2 ⋅ 6 x + 2 3 x ⋅ 6 x = − 3 x 3 + 4 x 2 \left( { - \frac{1}{2}{x^2} + \frac{2}{3}x} \right) \cdot 6x = - \frac{1}{2}{x^2} \cdot 6x + \frac{2}{3}x \cdot 6x = - 3{x^3} + 4{x^2} ( − 2 1 x 2 + 3 2 x ) ⋅ 6 x = − 2 1 x 2 ⋅ 6 x + 3 2 x ⋅ 6 x = − 3 x 3 + 4 x 2
г) 1 , 2 m 3 ⋅ ( 4 1 6 m 2 + 5 6 m − 2 11 12 ) = 1 , 2 m 3 ⋅ 4 1 6 m 2 + 1 , 2 m 3 ⋅ 5 6 m − 1 , 2 m 3 ⋅ 2 11 12 = 1,2{m^3} \cdot \left( {4\frac{1}{6}{m^2} + \frac{5}{6}m - 2\frac{{11}}{{12}}} \right) = 1,2{m^3} \cdot 4\frac{1}{6}{m^2} + 1,2{m^3} \cdot \frac{5}{6}m - 1,2{m^3} \cdot 2\frac{{11}}{{12}} = 1 , 2 m 3 ⋅ ( 4 6 1 m 2 + 6 5 m − 2 12 11 ) = 1 , 2 m 3 ⋅ 4 6 1 m 2 + 1 , 2 m 3 ⋅ 6 5 m − 1 , 2 m 3 ⋅ 2 12 11 =
= 12 10 m 3 ⋅ 25 6 m 2 + 12 10 m 3 ⋅ 5 6 m − 12 10 m 3 ⋅ 35 12 = 5 m 5 + m 4 − 3 , 5 m 3 = \frac{{12}}{{10}}{m^3} \cdot \frac{{25}}{6}{m^2} + \frac{{12}}{{10}}{m^3} \cdot \frac{5}{6}m - \frac{{12}}{{10}}{m^3} \cdot \frac{{35}}{{12}} = 5{m^5} + {m^4} - 3,5{m^3} = 10 12 m 3 ⋅ 6 25 m 2 + 10 12 m 3 ⋅ 6 5 m − 10 12 m 3 ⋅ 12 35 = 5 m 5 + m 4 − 3 , 5 m 3
Пр.2) Упростити изразе:
а) ( − 3 x 3 ) 2 ⋅ ( x − 3 ) = 9 x 6 ⋅ ( x − 3 ) = 9 x 6 ⋅ x − 9 x 6 ⋅ 3 = 9 x 7 − 27 x 6 {\left( { - 3{x^3}} \right)^2} \cdot \left( {x - 3} \right) = 9{x^6} \cdot \left( {x - 3} \right) = 9{x^6} \cdot x - 9{x^6} \cdot 3 = 9{x^7} - 27{x^6} ( − 3 x 3 ) 2 ⋅ ( x − 3 ) = 9 x 6 ⋅ ( x − 3 ) = 9 x 6 ⋅ x − 9 x 6 ⋅ 3 = 9 x 7 − 27 x 6
б) ( 8 x 2 + 16 x − 4 ) ⋅ ( − 1 2 x 2 ) 3 = 8 x 2 ⋅ ( − 1 8 x 6 ) + 16 x ⋅ ( − 1 8 x 6 ) − 4 ⋅ ( − 1 8 x 6 ) = \left( {8{x^2} + 16x - 4} \right) \cdot {\left( { - \frac{1}{2}{x^2}} \right)^3} = 8{x^2} \cdot \left( { - \frac{1}{8}{x^6}} \right) + 16x \cdot \left( { - \frac{1}{8}{x^6}} \right) - 4 \cdot \left( { - \frac{1}{8}{x^6}} \right) = ( 8 x 2 + 16 x − 4 ) ⋅ ( − 2 1 x 2 ) 3 = 8 x 2 ⋅ ( − 8 1 x 6 ) + 16 x ⋅ ( − 8 1 x 6 ) − 4 ⋅ ( − 8 1 x 6 ) =
= − x 8 − 2 x 7 + 1 2 x 6 = - {x^8} - 2{x^7} + \frac{1}{2}{x^6} = − x 8 − 2 x 7 + 2 1 x 6
в ( 0 , 1 x y ) 2 ⋅ ( 8 x 2 + 16 y − 4 y 2 ) = 0 , 01 x 2 y 2 ⋅ 8 x 2 + 0 , 01 x 2 y 2 ⋅ 16 y − 0 , 01 x 2 y 2 ⋅ 4 y 2 = {\left( {0,1xy} \right)^2} \cdot \left( {8{x^2} + 16y - 4{y^2}} \right) = 0,01{x^2}{y^2} \cdot 8{x^2} + 0,01{x^2}{y^2} \cdot 16y - 0,01{x^2}{y^2} \cdot 4{y^2} = ( 0 , 1 x y ) 2 ⋅ ( 8 x 2 + 16 y − 4 y 2 ) = 0 , 01 x 2 y 2 ⋅ 8 x 2 + 0 , 01 x 2 y 2 ⋅ 16 y − 0 , 01 x 2 y 2 ⋅ 4 y 2 = = 0 , 08 x 4 y 2 + 0 , 16 x 2 y 3 − 0 , 04 x 2 y 4 = 0,08{x^4}{y^2} + 0,16{x^2}{y^3} - 0,04{x^2}{y^4} = 0 , 08 x 4 y 2 + 0 , 16 x 2 y 3 − 0 , 04 x 2 y 4 Пр.3) а) x ⋅ ( 3 x 2 + 7 x − 10 ) + 3 x ⋅ ( 4 x 2 + 2 x − 1 ) = = 3 x 3 + 7 x 2 − 10 x + 12 x 3 + 6 x 2 − 3 x = = 15 x 3 + 13 x 2 − 13 x \begin{aligned}&x \cdot \left( {3{x^2} + 7x - 10} \right) + 3x \cdot \left( {4{x^2} + 2x - 1} \right) = \\ &= 3{x^3} + 7{x^2} - 10x + 12{x^3} + 6{x^2} - 3x = \\ &= 15{x^3} + 13{x^2} - 13x\end{aligned} x ⋅ ( 3 x 2 + 7 x − 10 ) + 3 x ⋅ ( 4 x 2 + 2 x − 1 ) = = 3 x 3 + 7 x 2 − 10 x + 12 x 3 + 6 x 2 − 3 x = = 15 x 3 + 13 x 2 − 13 x
б)
− 3 y 2 ⋅ ( 2 y 2 + 3 y + 4 ) − 6 y ⋅ ( y 3 − 2 y 2 + 7 y ) = = − 6 y 4 − 9 y 3 − 12 y 2 − 6 y 4 + 12 y 3 − 42 y 2 = = − 12 y 4 + 3 y 3 − 54 y 2 \begin{aligned}&- 3{y^2} \cdot \left( {2{y^2} + 3y + 4} \right) - 6y \cdot \left( {{y^3} - 2{y^2} + 7y} \right) = \\ &= - 6{y^4} - 9{y^3} - 12{y^2} - 6{y^4} + 12{y^3} - 42{y^2} = \\ &= - 12{y^4} + 3{y^3} - 54{y^2}\end{aligned} − 3 y 2 ⋅ ( 2 y 2 + 3 y + 4 ) − 6 y ⋅ ( y 3 − 2 y 2 + 7 y ) = = − 6 y 4 − 9 y 3 − 12 y 2 − 6 y 4 + 12 y 3 − 42 y 2 = = − 12 y 4 + 3 y 3 − 54 y 2
в) ( − 1 3 x 2 + 1 2 x ) ⋅ 6 x + 4 x 2 ⋅ ( − 1 2 x + 3 4 ) = = − 2 x 3 + 3 x 2 − 2 x 3 + 3 x 2 = − 4 x 3 + 6 x 2 \begin{aligned}&\left( { - \frac{1}{3}{x^2} + \frac{1}{2}x} \right) \cdot 6x + 4{x^2} \cdot \left( { - \frac{1}{2}x + \frac{3}{4}} \right) = \\ &= - 2{x^3} + 3{x^2} - 2{x^3} + 3{x^2} = - 4{x^3} + 6{x^2}\end{aligned} ( − 3 1 x 2 + 2 1 x ) ⋅ 6 x + 4 x 2 ⋅ ( − 2 1 x + 4 3 ) = = − 2 x 3 + 3 x 2 − 2 x 3 + 3 x 2 = − 4 x 3 + 6 x 2
Пр.4) ( 3 x ) 2 ⋅ ( 2 x − 5 ) − x 2 ( 4 x − 3 ) = 9 x 2 ⋅ ( 2 x − 5 ) − x 2 ( 4 x − 3 ) = = 18 x 3 − 45 x 2 − 4 x 3 + 3 x 2 = 14 x 3 − 42 x 2 \begin{aligned}&{\left( {3x} \right)^2} \cdot \left( {2x - 5} \right) - {x^2}\left( {4x - 3} \right) = 9{x^2} \cdot \left( {2x - 5} \right) - {x^2}\left( {4x - 3} \right) = \\ &= 18{x^3} - 45{x^2} - 4{x^3} + 3{x^2} = 14{x^3} - 42{x^2}\end{aligned} ( 3 x ) 2 ⋅ ( 2 x − 5 ) − x 2 ( 4 x − 3 ) = 9 x 2 ⋅ ( 2 x − 5 ) − x 2 ( 4 x − 3 ) = = 18 x 3 − 45 x 2 − 4 x 3 + 3 x 2 = 14 x 3 − 42 x 2
У к р а т к о Моном множите полиномом тако што моном помножите сваким чланом полинома, па добијене производе саберете: a ( b + c − d ) = a b + a c − a d a(b + c - d) = ab + ac - ad a ( b + c − d ) = ab + a c − a d 3 x ( 2 x 2 − 5 x + 4 ) = 6 x 3 − 15 x 2 + 12 x 3x(2x^2 - 5x + 4) = 6x^3 - 15x^2 + 12x 3 x ( 2 x 2 − 5 x + 4 ) = 6 x 3 − 15 x 2 + 12 x
Пазите на знакове: кад је моном негативан, мења се знак сваком члану у загради, − 2 x ( x − 3 ) = − 2 x 2 + 6 x -2x(x - 3) = -2x^2 + 6x − 2 x ( x − 3 ) = − 2 x 2 + 6 x ( − 3 x 2 ) 2 ( x − 1 ) = 9 x 4 ( x − 1 ) = 9 x 5 − 9 x 4 (-3x^2)^2 (x - 1) = 9x^4(x - 1) = 9x^5 - 9x^4 ( − 3 x 2 ) 2 ( x − 1 ) = 9 x 4 ( x − 1 ) = 9 x 5 − 9 x 4 x = 1 x = 1 x = 1
М е њ а ј т е ш и р и н у x x² = 32412x = 216 x = 18x = 1812 Ширина x 18 m
Квадрат x² 324 m²
Трака 12x 216 m²
Површина x(x + 12) 540 m² Плац x x x x + 12 x + 12 x + 12 x 2 x^2 x 2 12 x 12x 12 x x ( x + 12 ) x(x + 12) x ( x + 12 ) П р и м е р и з ж и в о т а Дистрибутивни закон користи се стално: у рачунању напамет, као 7 ⋅ 104 = 7 ⋅ 100 + 7 ⋅ 4 = 728 7 \cdot 104 = 7 \cdot 100 + 7 \cdot 4 = 728 7 ⋅ 104 = 7 ⋅ 100 + 7 ⋅ 4 = 728
Површина плаца Правоугаони плац широк је x x x x = 18 x = 18 x = 18
Решење Дужина плаца је x + 12 x + 12 x + 12 P = x ( x + 12 ) P = x(x + 12) P = x ( x + 12 ) Моном x x x P = x ⋅ x + x ⋅ 12 = x 2 + 12 x P = x \cdot x + x \cdot 12 = x^2 + 12x P = x ⋅ x + x ⋅ 12 = x 2 + 12 x За x = 18 x = 18 x = 18 P = 18 2 + 12 ⋅ 18 = 324 + 216 = 540 P = 18^2 + 12 \cdot 18 = 324 + 216 = 540 P = 1 8 2 + 12 ⋅ 18 = 324 + 216 = 540 Провера: плац је дугачак 18 + 12 = 30 18 + 12 = 30 18 + 12 = 30 18 ⋅ 30 = 540 18 \cdot 30 = 540 18 ⋅ 30 = 540 Одговор: Површина плаца је x 2 + 12 x x^2 + 12x x 2 + 12 x x = 18 x = 18 x = 18
Задатак за вежбу Помножите 2 x ( 3 x 2 − 4 x + 5 ) 2x(3x^2 - 4x + 5) 2 x ( 3 x 2 − 4 x + 5 ) x = 1 x = 1 x = 1
Прикажи решење 2 x ( 3 x 2 − 4 x + 5 ) = 6 x 3 − 8 x 2 + 10 x 2x(3x^2 - 4x + 5) = 6x^3 - 8x^2 + 10x 2 x ( 3 x 2 − 4 x + 5 ) = 6 x 3 − 8 x 2 + 10 x x = 1 x = 1 x = 1 2 ⋅ ( 3 − 4 + 5 ) = 8 2 \cdot (3 - 4 + 5) = 8 2 ⋅ ( 3 − 4 + 5 ) = 8 6 − 8 + 10 = 8 6 - 8 + 10 = 8 6 − 8 + 10 = 8
