Operasi Bentuk Aljabar

Penjumlahan dan Pengurangan

Penjumlahan dan pengurangan pada bentuk aljabar hanya dapat dilakukan pada suku sejenis.

Contoh 1:

Tentukan hasil operasi penjumlahan atau pengurangan berikut ini!

1. $\left( {2x + 3y - 4} \right) + \left( {4x - 5y + 7} \right)$
2. $\left( {p + 3q - 6} \right) - \left( {3p - 4q + 7} \right)$
3. $2{x^2} + 3x - 5 + {x^2} + 4x - 6$
4. $\begin{array}{l} 3a + 5b\\ \underline {4a - 6b} + \end{array}$


5. $\begin{array}{l} 2a - 5b\\ \underline {4a - 4b} - \end{array}$

Jawab:

1. Soal 1
$\begin{array}{l} \left( {2x + 3y - 4} \right) + \left( {4x - 5y + 7} \right)\\ = 2x + 3y - 4 + 4x - 5y + 7\\ = 2x + 4x + 3y - 5y - 4 + 7\\ = \left( {2 + 4} \right)x + \left( {3 - 5} \right)y + \left( { - 4 + 7} \right)\\ = 6x + \left( { - 2} \right)y + 3\\ = 6x - 2y + 3 \end{array}$
2. Soal 2
$\begin{array}{l} \left( {p + 3q - 6} \right) - \left( {3p - 4q + 7} \right)\\ = p + 3q - 6 - 3p + 4q - 7\\ = p - 3p + 3q + 4q - 6 - 7\\ = \left( {1 - 3} \right)p + \left( {3 + 4} \right)q + \left( { - 6 - 7} \right)\\ = - 2p + 7q - 13 \end{array}$
3. Soal 3
$\begin{array}{l} 2{x^2} + 3x - 5 + {x^2} + 4x - 6\\ = 2{x^2} + {x^2} + 3x + 4x - 5 - 6\\ = \left( {2 + 1} \right){x^2} + \left( {3 + 4} \right)x + \left( { - 5 - 6} \right)\\ = 3{x^2} + 7x - 11 \end{array}$
4. Soal 4
$\begin{array}{l} \begin{array}{*{20}{l}} {3a + 5b}\\ {\underline {4a - 6b} + } \end{array}\\ 7a - b \end{array}$


5. Soal 5
$\begin{array}{l} 2a - 5b\\ \underline {4a - 4b} - \\ 6a - b \end{array}$

Perkalian

Rumus yang perlu diperhatikan untuk perkalian bentuk aljabar adalah $\boxed{{x^a} \times {x^b} = {x^{a + b}}}$

Contoh 2:

Tentukan hasil operasi perkalian berikut ini!

1. $2x \times 3x$
2. $2{x^2} \times 4{x^3}$
3. $ - 3x{y^2} \times 2{x^3}yz$
4. $2x\left( {3x + 5} \right)$
5. $3x\left( {{x^2} - 4x + 1} \right)$
6. $\left( {x + 3} \right)\left( {x - 4} \right)$
7. $\left( {x + 2} \right)\left( {2{x^2} - x + 7} \right)$

Jawab:

1. $2x \times 3x = 6{x^2}\left( {\begin{array}{*{20}{c}} {2 \times 3 = 6}&{x \times x = {x^2}} \end{array}} \right)$
2. $2{x^2} \times 4{x^3} = 8{x^5}\left( {\begin{array}{*{20}{c}} {2 \times 4 = 8,}&{{x^2} \times {x^3} = {x^5}} \end{array}} \right)$
3. $ - 3x{y^2} \times 2{x^3}yz = - 6{x^4}{y^3}z\left( {\begin{array}{*{20}{c}} { - 3 \times 2 = - 6,}&{x \times {x^3} = {x^4},}&{{y^2} \times y = {y^3},}&{1 \times z = z} \end{array}} \right)$
4. $2x\left( {3x + 5} \right) = 6{x^2} + 10x\left( {\begin{array}{*{20}{c}} {2x \times 3x = 6{x^2}}&{2x \times 5 = 10x} \end{array}} \right)$
5. $3x\left( {{x^2} - 4x + 1} \right) = 3{x^3} - 12{x^2} + 3x\left( {\begin{array}{*{20}{c}} {3x \times {x^2} = 3{x^3}}&{3x \times \left( { - 4x} \right) = - 12x}&{3x \times 1 = 3x} \end{array}} \right)$
6. $\begin{array}{l} \left( {x + 3} \right)\left( {x - 4} \right)\\ = x\left( {x - 4} \right) + 3\left( {x - 4} \right)\\ = {x^2} - 4x + 3x - 12\\ = {x^2} - x - 12 \end{array}$
7. $\begin{array}{l} \left( {x + 2} \right)\left( {2{x^2} - x + 7} \right)\\ = x\left( {2{x^2} - x + 7} \right) + 2\left( {2{x^2} - x + 7} \right)\\ = 2{x^3} - {x^2} + 7x + 4{x^2} - 2x + 14\\ = 2{x^3} - {x^2} + 4{x^2} + 7x - 2x + 14\\ = 2{x^3} + 3{x^2} + 5x + 14 \end{array}$

Pembagian

Rumus yang perlu diperhatikan untuk pembagian bentuk aljabar adalah $\boxed{{x^a}:{x^b} = {x^{a - b}}}$

Contoh 3:

Tentukan hasil operasi pembagian berikut ini!

1. $4x:x$
2. $9{x^3}:3x$
3. $12{x^4}{y^3}: - 6x{y^2}$
4. $\left( {2x + 4} \right):2$
5. $\left( {3{x^2} - 6x} \right):3x$
6. $\left( {{x^2} + 3x + 2} \right):\left( {x + 1} \right)$
7. $\left( {2{x^3} + 3{x^2} + 5x + 14} \right):\left( {x + 2} \right)$

Jawab:

1. $4x:x = 4\left( {\begin{array}{*{20}{c}} {4:1 = 4}&{x:x = 1} \end{array}} \right)$
2. $9{x^3}:3x = 3{x^2}\left( {\begin{array}{*{20}{c}} {9:3 = 3}&{{x^3}:x = {x^2}} \end{array}} \right)$
3. $12{x^4}{y^3}: - 6x{y^2} = - 2{x^3}y\left( {\begin{array}{*{20}{c}} {12:\left( { - 6} \right) = - 2}&{{x^4}:x = {x^3}}&{{y^3}:{y^2} = y} \end{array}} \right)$
4. $\begin{array}{l} \left( {2x + 4} \right):2\\ = \frac{{2x + 4}}{2} = \frac{{2x}}{2} + \frac{4}{2} = x + 2 \end{array}$
5. $\begin{array}{l} \left( {3{x^2} - 6x} \right):3x\\ = \frac{{3{x^2} - 6x}}{{3x}} = \frac{{3{x^2}}}{{3x}} - \frac{{6x}}{{3x}}\\ = x - 2 \end{array}$
6. Soal 6
   $\left( {{x^2} + 3x + 2} \right):\left( {x + 1} \right) = $
   $\begin{array}{l} \begin{array}{*{20}{c}} {}&{}&{}&{x + 2} \end{array}\\ x + 1\sqrt {{x^2} + 3x + 2} \\ \begin{array}{*{20}{c}} {}&{}&\begin{array}{l} \underline {{x^2} + x} - \\ \begin{array}{*{20}{c}} {}&\begin{array}{l} 2x + 2\\ \underline {2x + 2} - \\ \begin{array}{*{20}{c}} {}&0 \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}$
7. Soal 7
   $\left( {2{x^3} + 3{x^2} + 5x + 14} \right):\left( {x + 2} \right) = $
   $\begin{array}{l} \begin{array}{*{20}{c}} {}&{}&{2{x^2} - x + 7} \end{array}\\ x + 2\left){\vphantom{1{2{x^3} + 3{x^2} + 5x + 14}}}\right. \!\!\!\!\overline{\,\,\,\vphantom 1{{2{x^3} + 3{x^2} + 5x + 14}}}\\ \begin{array}{*{20}{c}} {}&{}&\begin{array}{l} \underline {2{x^3} + 4{x^2}} - \\ \begin{array}{*{20}{c}} {}&\begin{array}{l} - {x^2} + 5x\\ \underline { - {x^2} - 2x} - \\ \begin{array}{*{20}{c}} {}&{}&\begin{array}{l} 7x + 14\\ \underline {7x + 14} - \\ \begin{array}{*{20}{c}} {}&0 \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}$

Perpangkatan

Rumus yang perlu diperhatikan untuk perpangkatan bentuk aljabar adalah $\boxed{{{\left( {{x^a}} \right)}^b} = {x^{a \times b}}}$

Contoh 4:

Tentukan hasil operasi perpangkatan bentuk aljabar berikut ini!

1. ${\left( {2x} \right)^3}$
2. ${\left( {3{x^2}y{z^3}} \right)^2}$
3. ${\left( {x + 2} \right)^2}$
4. ${\left( {2x - 3} \right)^3}$

Jawab:

1. ${\left( {2x} \right)^3} = {2^3} \times {x^3} = 8 \times {x^3} = 8{x^3}$
2. ${\left( {3{x^2}y{z^3}} \right)^2} = {3^2}.{\left( {{x^2}} \right)^2}.{y^2}.{\left( {{z^3}} \right)^2} = 9.{x^4}.{y^2}.{z^6} = 9{x^4}{y^2}{z^6}$
3. $\begin{array}{l} {\left( {x + 2} \right)^2} = \left( {x + 2} \right)\left( {x + 2} \right)\\ = x\left( {x + 2} \right) + 2\left( {x + 2} \right)\\ = {x^2} + 2x + 2x + 4\\ = {x^2} + 4x + 4 \end{array}$
4. $\begin{array}{l} {\left( {2x - 3} \right)^3} = \left( {2x - 3} \right)\left( {2x - 3} \right)\left( {2x - 3} \right)\\ = \left( {2x - 3} \right)\left[ {2x\left( {2x - 3} \right) - 3\left( {2x - 3} \right)} \right]\\ = \left( {2x - 3} \right)\left[ {4{x^2} - 6x - 6x + 9} \right]\\ = \left( {2x - 3} \right)\left( {4{x^2} - 12x + 9} \right)\\ = 2x\left( {4{x^2} - 12x + 9} \right) - 3\left( {4{x^2} - 12x + 9} \right)\\ = 8{x^3} - 24{x^2} + 18x - 12{x^2} + 36x - 27\\ = 8{x^3} - 24{x^2} - 12{x^2} + 18x + 36x - 27\\ = 8{x^3} - 36{x^2} + 54x - 27 \end{array}$