Pemfaktoran dan Penjabaran Bentuk a² − b²
\(\color{blue} a^2\: – \:b^2 = (a + b)(a\: – \:b)\)
Soal Latihan
Soal 1
Faktorkan \(x^2\: – \:9\)
\begin{equation*}
\begin{split}
x^2\: -\: 9& = x^2 \:- \:3^2\\\\
x^2 \:- \:9& = (x + 3)(x\: – \:3)
\end{split}
\end{equation*}
Soal 2
Faktorkan \(x^2\: – \:\dfrac{1}{36}\)
\begin{equation*}
\begin{split}
x^2\: – \:\frac{1}{36}& = x^2 \:- \:\left(\frac{1}{6}\right)^2\\\\
x^2\: – \:\frac{1}{36}& = (x + \frac{1}{6})(x\: – \:\frac{1}{6})
\end{split}
\end{equation*}
Soal 3
Faktorkan \(1\: -\: x^2\)
\begin{equation*}
\begin{split}
1\: – \:x^2& = 1^2\: – \:x^2\\\\
1\: – \:x^2& = (1 + x)(1\:- \:x)
\end{split}
\end{equation*}
Soal 4
Faktorkan \(x^2 \:- \:121y^2\)
\begin{equation*}
\begin{split}
x^2\: – \:121y^2& = x^2\: – \:11^2y^2\\\\
x^2\: – \:121y^2& = x^2 \:- \:(11y)^2\\\\
x^2\: – \:121y^2& = (x + 11y)(x\: – \:11y)
\end{split}
\end{equation*}
Soal 5
Faktorkan \(4x^2 \:- \:16y^2\)
\begin{equation*}
\begin{split}
4x^2\: – \:16y^2& = 2^2x^2 \:- \:4^2y^2\\\\
4x^2 \:- \:16y^2& = (2x)^2\: – \:(4y)^2\\\\
4x^2 \:- \:16y^2& = (2x + 4y)(2x \:- \:4y)
\end{split}
\end{equation*}
Soal 6
Faktorkan \(m^2n^2 \:-\: 81\)
\begin{equation*}
\begin{split}
m^2n^2 \:-\: 81& = (mn)^2 \:- \:9^2\\\\
m^2n^2 \:-\: 81& = (mn + 9)(mn\: -\: 9)
\end{split}
\end{equation*}
Soal 7
Jabarkan \((2x + 7)(2x\: – \:7)\)
\begin{equation*}
\begin{split}
(2x + 7)(2x\: – \:7)& = (2x)^2\: – \:7^2\\\\
(2x + 7)(2x\: – \:7)& = 4x^2 \:- \:49
\end{split}
\end{equation*}
Soal 8
Jabarkan \((m^2 + 12)(m^2\: – \:12)\)
\begin{equation*}
\begin{split}
(m^2 + 12)(m^2\: – \:12)& = (m^2)^2\: -\: 12^2\\\\
(m^2 + 12)(m^2\: – \:12)& = m^4 \:- \:144
\end{split}
\end{equation*}
Soal 9
Jabarkan \(\left(2m + \dfrac{3}{5}\right)\left(2m\:-\: \dfrac{3}{5}\right)\)
\begin{equation*}
\begin{split}
\left(2m + \frac{3}{5}\right)\left(2m\:- \:\frac{3}{5}\right)& = (2m)^2 \:-\: \left(\frac{3}{5}\right)^2\\\\
\left(2m + \frac{3}{5}\right)\left(2m\:- \:\frac{3}{5}\right)& =2^2m^2 \:-\: \left(\frac{3^2}{5^2}\right)\\\\
\left(2m + \frac{3}{5}\right)\left(2m\:- \:\frac{3}{5}\right)& = 4m^2 \:-\:\left(\frac{9}{25}\right)
\end{split}
\end{equation*}
Soal 10
Jabarkan \((2x + 5y^3)(2x \:- \:5y^3)\)
\begin{equation*}
\begin{split}
(2x + 5y^3)(2x \:- \:5y^3)& = (2x)^2 \:- \:(5y^3)^2\\\\
(2x + 5y^3)(2x \:- \:5y^3)& = 2^2x^2 \:- \:5^2y^6\\\\
(2x + 5y^3)(2x \:- \:5y^3)& = 4x^2\: -\: 25y^6
\end{split}
\end{equation*}
Soal 11
Jabarkan \((p^5 + 2mn)(p^5 \:- \:2mn)\)
\begin{equation*}
\begin{split}
(p^5 + 2mn)(p^5 \:- \:2mn)& = (p^5)^2\: -\: (2mn)^2\\\\
(p^5 + 2mn)(p^5 \:- \:2mn)& = p^{10} \:- \:2^2m^2n^2\\\\
(p^5 + 2mn)(p^5 \:- \:2mn)& = p^{10} \:- \:4m^2n^2
\end{split}
\end{equation*}
Soal 12
Jabarkan \((5x^3 + y^7)(5x^3\: – \:y^7)\)
\begin{equation*}
\begin{split}
(5x^3 + y^7)(5x^3\: – \:y^7)& = (5x^3)^2\: -\: (y^7)^2\:\:\:\:\color{blue}(a^m)^n = a^{m\times n}\\\\
(5x^3 + y^7)(5x^3\: – \:y^7)& = 5^2x^{3\times 2}\: -\: y^{7\times 2}\\\\
(5x^3 + y^7)(5x^3\: – \:y^7)& = 25x^6 \: – \:y^{14}
\end{split}
\end{equation*}
Soal 13
Faktorkan \(-12m^2 + 27n^2\)
\begin{equation*}
\begin{split}
-12m^2 + 27n^2& = -3(4m^2 \:- \:9n^2)\\\\
-12m^2 + 27n^2& = -3(2^2m^2\: – \:3^2n^2)\\\\
-12m^2 + 27n^2& = -3[(2m)^2 \:- \:(3n)^2]\\\\
-12m^2 + 27n^2& = -3(2m + 3n)(2m \:-\: 3n)
\end{split}
\end{equation*}
Soal 14
Faktorkan \(36m^4n^2\: -\: 5\)
\begin{equation*}
\begin{split}
36m^4n^2 \: -\: 5& = 6^2(m^2)^2n^2\: – \:(\sqrt{5})^2\\\\
36m^4n^2 \: -\: 5& = (6m^2n)^2\: – \: (\sqrt{5})^2\\\\
36m^4n^2 \: -\: 5& = (6m^2n + \sqrt{5})(6m^2n \:-\: \sqrt{5})
\end{split}
\end{equation*}
Soal 15
Faktorkan \(\dfrac{49}{121}x^4\: -\: \dfrac{25}{64}\)
\begin{equation*}
\begin{split}
\dfrac{49}{121}x^2 \:- \:\dfrac{25}{64}& = \dfrac{7^2}{11^2}x^2 \:-\: \dfrac{5^2}{8^2}\\\\
\dfrac{49}{121}x^2 \:- \:\dfrac{25}{64}& = \left(\dfrac{7}{11}x\right)^2 – \left(\dfrac{5}{8}\right)^2\\\\
\dfrac{49}{121}x^2 \:- \:\dfrac{25}{64}& = \left(\dfrac{7}{11}x +\dfrac{5}{8}\right)\left(\dfrac{7}{11}x – \dfrac{5}{8}\right)
\end{split}
\end{equation*}
Soal 16
Jabarkan \((x + 9)(x\: – \:9)(x^2 + 81)\)
\begin{equation*}
\begin{split}
(x + 9)(x\: – \:9)(x^2 + 81)& = (x^2 \:- \:9^2)(x^2 + 81)\\\\
(x + 9)(x\: – \:9)(x^2 + 81)& = (x^2\: – \:81)(x^2 + 81)\\\\
(x + 9)(x\: – \:9)(x^2 + 81)& = (x^2)^2\: – \:81^2\\\\
(x + 9)(x\: – \:9)(x^2 + 81)& = x^4 \:- \:6561
\end{split}
\end{equation*}
Soal 17
Jabarkan \((m + 2n)(m – 2n)(m^2 + 4n^2)\)
\begin{equation*}
\begin{split}
(m + 2n)(m – 2n)(m^2 + 4n^2)& = [m^2 \:-\: (2n)^2](m^2 + 4n^2)\\\\
(m + 2n)(m – 2n)(m^2 + 4n^2)& = (m^2 \:- \:4n^2)(m^2 + 4n^2)\\\\
(m + 2n)(m – 2n)(m^2 + 4n^2)& = (m^2)^2\: -\: (4n^2)^2\\\\
(m + 2n)(m – 2n)(m^2 + 4n^2)& = m^4 \:- \:4^2n^4\\\\
(m + 2n)(m – 2n)(m^2 + 4n^2)& = m^4\: -\: 16n^4
\end{split}
\end{equation*}
Soal 18
Jabarkan \((-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)\)
\begin{equation*}
\begin{split}
(-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)& = -(2xy + 1)(2xy\: -\: 1)(4x^2y^2 + 1)\\\\
(-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)& = -[(2xy)^2\: -\: 1^2](4x^2y^2 + 1)\\\\
(-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)& = -(4x^2y^2 \:-\: 1)(4x^2y^2 + 1)\\\\
(-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)& = -[(4x^2y^2)^2 \:- \:1^2]\\\\
(-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)& = -(16x^4y^4\: – \:1)\\\\
(-2xy\: – \:1)(2xy\: -\: 1)(4x^2y^2 + 1)& = -16x^4y^4 + 1
\end{split}
\end{equation*}
Soal 19
Faktorkan \(256m^8 \: – \: n^8\)
\begin{equation*}
\begin{split}
&2^8m^8 \: – \:n^8\\\\
&(2^4m^4)^2 \: -\: (n^4)^2\\\\
&(2^4m^4 + n^4)(2^4m^4 \:- \:n^4)\\\\
&(16m^4 + n^4)[(2^2m^2)^2 \:-\: (n^2)^2]\\\\
&(16m^4 + n^4)(2^2m^2 + n^2)(2^2m^2\: -\: n^2)\\\\
&(16m^4 + n^4)(4m^2 + n^2)[(2m)^2 \:- \:n^2]\\\\
&(16m^4 + n^4)(4m^2 + n^2)(2m + n)(2m \:-\: n)\\\\
\end{split}
\end{equation*}
Soal 20
Faktorkan \((x + y\: – \:9)^2\: – \:(x + y + 9)^2\)
\begin{equation*}
\begin{split}
&(x + y \:-\: 9)^2 \:- \:(x + y + 9)^2\\\\
&(x + y \:-\: 9 + x + y +9)[x + y\: – \:9\: -\: (x + y + 9)]\\\\
&(2x +2y)(x + y\: -\: 9 \:- \:x\: -\: y\: -\: 9)\\\\
&2(x + y)(-18)\\\\
&-36(x + y)
\end{split}
\end{equation*}
Soal 21
Faktorkan \((\sqrt{2} + 5mn)^4 \:-\: (\sqrt{2}\:- \:5mn)^4\)
\begin{equation*}
\begin{split}
&(\sqrt{2} + 5mn)^4 \:- \:(\sqrt{2}\: -\: 5mn)^4\\\\
&[(\sqrt{2} + 5mn)^2]^2 \:- \:[(\sqrt{2}\: – \:5mn)^2]^2\\\\
&[(\sqrt{2} + 5mn)^2 +(\sqrt{2}\: – \:5mn)^2][(\sqrt{2} + 5mn)^2 \:-\:(\sqrt{2}\: – \:5mn)^2 ]\:\:\:\:\:\color{blue} (a \pm b)^2 = a^2 \pm 2ab + b^2\\\\
&[(\sqrt{2})^2 + \cancel{2\cdot\sqrt{2}\cdot 5mn} + (5mn)^2 + (\sqrt{2})^2\: – \:\cancel{2\cdot\sqrt{2}\cdot 5mn} + (5mn)^2][\cancel{(\sqrt{2})^2} + 2\cdot\sqrt{2}\cdot 5mn + \cancel{(5mn)^2} – [\cancel{(\sqrt{2})^2} \:-\: 2\cdot\sqrt{2}\cdot 5mn + \cancel{(5mn)^2}]]\\\\
&(2 + 25m^2n^2 + 2 + 25m^2n^2)(10\sqrt{2}mn + 10\sqrt{2}mn )\\\\
&(4 + 50m^2n^2)(20\sqrt{2}mn)\\\\
&2(2 + 25m^2n^2)(20\sqrt{2}mn)\\\\
&40\sqrt{2}mn(2 + 25m^2n^2)
\end{split}
\end{equation*}
Soal 22
Jabarkan \((2x + y + \sqrt{2} + 1)(2x + y\: – \:\sqrt{2}\:-\: 1)\)
\begin{equation*}
\begin{split}
&(2x + y + \sqrt{2} + 1)(2x + y \:-\: \sqrt{2}\: – \:1)\\\\\
&[(2x + y) + (\sqrt{2} + 1)][(2x + y) \:-\: (\sqrt{2} + 1)]\\\\
&(2x + y)^2 – (\sqrt{2} + 1)^2\:\:\:\:\:\color{blue} (a + b)^2 = a^2 + 2ab + b^2\\\\
&(2x)^2 + 2\cdot 2x \cdot y + y^2\: – \:[(\sqrt{2})^2 + 2\cdot \sqrt{2}\cdot 1 + 1^2]\\\\
&4x^2 + 4xy + y^2 \: – \:(2 + 2\sqrt{2} + 1)\\\\
&4x^2 + 4xy + y^2 \:- \:2 \:- \:2\sqrt{2} \:- \:1\\\\
&4x^2 + 4xy + y^2 \:- \:2\sqrt{2}\:-\: 3
\end{split}
\end{equation*}
Soal 23
Jabarkan \((\sqrt{2} + x + y)(\sqrt{2}\: -\: x \:- \:y)(2 + x^2 + 2xy + y^2)\)
\begin{equation*}
\begin{split}
&(\sqrt{2} + x + y)(\sqrt{2}\: -\: x\: -\: y)(2 + x^2 + 2xy + y^2)\\\\
&[\sqrt{2} + (x + y)][\sqrt{2}\: -\: (x + y)](2 + (x + y)^2)\\\\
&[(\sqrt{2})^2\: – \:(x + y)^2][(\sqrt{2})^2 + (x + y)^2]\\\\
&[(\sqrt{2})^4 \:-\: (x + y)^4]\\\\
&4\: -\: (x + y)^4\:\:\:\:\:\color{blue} (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\\\\
&4 \:-\: (x^4 +4x^3y + 6x^2y^2 + 4xy^3 + y^4)\\\\
&-x^4 \:-\: 4x^3y\: – \:6x^2y^2\: -\: 4xy^3 \:-\: y^4 + 4
\end{split}
\end{equation*}
Soal 24
Jabarkan \((m^8 \:- \:n^8 + r^8)(m^8 + n^8 \:- \:r^8)\)
\begin{equation*}
\begin{split}
&(m^8\: -\: n^8 + r^8)(m^8 + n^8 \:-\: r^8)\\\\
&[m^8 \:- \:(n^8\: -\: r^8)][m^8 + (n^8\: -\: r^8)]\\\\
&(m^8)^2\: – \:(n^8 \:-\: r^8)^2\:\:\:\:\:\color{blue} (a \:- \:b)^2 = a^2 \:- \:2ab + b^2\\\\
&m^{16}\:-\:[(n^8)^2 \:-\: 2\cdot n^8 \cdot r^8 + (r^8)^2]\\\\
&m^{16}\: -\: (n^{16}\: -\: 2n^8r^8 + r^{16})\\\\
&m^{16} \:-\: n^{16} + 2n^8r^8 \:- \:r^{16}
\end{split}
\end{equation*}
