Soalan 3:
Faktorkan ungkapan algebra berikut.
(a) b2 – 81
(b) a2 – b2
(c) x2 – 1
(d) 16y2 – 49
(e) (m + 3)2 – 16
(f) 4(x – 1)2 – 9
Penyelesaian:
Menggunakan beza antara dua sebutan kuasa dua sempurna
x2 – y2 ialah sebutan beza kuasa dua. x2 – y2 boleh difaktorkan dengan beza kuasa dua sempurna. Kaedah ini hanya boleh digunakan jika kedua-dua sebutan algebra tersebut ialah kuasa dua sempurna.
(a)
$$ \begin{aligned} & b^2-81 \\ & =b^2-9^2 \\ & =(b+9)(b-9) \end{aligned} $$
(b)
$$ \begin{aligned} & a^2-b^2 \\ & =a^2-b^2 \\ & =(a+b)(a-b) \end{aligned} $$
(c)
$$ \begin{aligned} & x^2-1 \\ & =x^2-1^2 \\ & =(x+1)(x-1) \end{aligned} $$
(d)
$$ \begin{aligned} & 16 y^2-49 \\ & =(4 y)^2-7^2 \\ & =(4 y+7)(4 y-7) \end{aligned} $$
(e)
$$ \begin{aligned} & (m+3)^2-16 \\ & =(m+3)^2-4^2 \\ & =(m+3+4)(m+3-4) \\ & =(m+7)(m-1) \end{aligned} $$
(f)
$$ \begin{aligned} & 4(x-1)^2-9 \\ & =2^2(x-1)^2-3^2 \\ & =[2(x-1)]^2-3^2 \\ & =[2(x-1)+3][2(x-1)-3] \\ & =(2 x-2+3)(2 x-2-3) \\ & =(2 x+1)(2 x-5) \end{aligned} $$
Faktorkan ungkapan algebra berikut.
(a) b2 – 81
(b) a2 – b2
(c) x2 – 1
(d) 16y2 – 49
(e) (m + 3)2 – 16
(f) 4(x – 1)2 – 9
Penyelesaian:
Menggunakan beza antara dua sebutan kuasa dua sempurna
x2 – y2 ialah sebutan beza kuasa dua. x2 – y2 boleh difaktorkan dengan beza kuasa dua sempurna. Kaedah ini hanya boleh digunakan jika kedua-dua sebutan algebra tersebut ialah kuasa dua sempurna.
(a)
$$ \begin{aligned} & b^2-81 \\ & =b^2-9^2 \\ & =(b+9)(b-9) \end{aligned} $$
(b)
$$ \begin{aligned} & a^2-b^2 \\ & =a^2-b^2 \\ & =(a+b)(a-b) \end{aligned} $$
(c)
$$ \begin{aligned} & x^2-1 \\ & =x^2-1^2 \\ & =(x+1)(x-1) \end{aligned} $$
(d)
$$ \begin{aligned} & 16 y^2-49 \\ & =(4 y)^2-7^2 \\ & =(4 y+7)(4 y-7) \end{aligned} $$
(e)
$$ \begin{aligned} & (m+3)^2-16 \\ & =(m+3)^2-4^2 \\ & =(m+3+4)(m+3-4) \\ & =(m+7)(m-1) \end{aligned} $$
(f)
$$ \begin{aligned} & 4(x-1)^2-9 \\ & =2^2(x-1)^2-3^2 \\ & =[2(x-1)]^2-3^2 \\ & =[2(x-1)+3][2(x-1)-3] \\ & =(2 x-2+3)(2 x-2-3) \\ & =(2 x+1)(2 x-5) \end{aligned} $$
Soalan 4:
Faktorkan ungkapan algebra berikut.
(a) x2 + 9x + 14
(b) x2 + 7x – 18
(c) x2 – 5x – 24
(d) m2 + 11m – 26
(e) y2 – 2y – 15
(f) k2 – 8k + 16
(g) 2m2 – 11m – 6
(h) 9f2 – 12f + 4
(i) 2m2 + 4m – 16
(j) 2x2 – 5x – 7
(k) 12y2 + 8y – 15
(l) 5p2 + 6p – 8
(m) –5m2 – 6m + 8
(n) –3p2 + 8p − 4
(o) –6x2 – x + 15
Penyelesaian:
(a)
$$ \begin{aligned} & x^2+9 x+14 \\ & =(x+7)(x+2) \end{aligned} $$
(b)
$$ \begin{aligned} & x^2+7 x-18 \\ & =(x+9)(x-2) \end{aligned} $$
(c)
$$ \begin{aligned} & x^2-5 x-24 \\ & =(x+3)(x-8) \end{aligned} $$
(d)
$$ \begin{aligned} & m^2+11 m-26 \\ & =(m+13)(m-2) \end{aligned} $$
(e)
$$ \begin{aligned} & y^2-2 y-15 \\ & =(y+3)(y-5) \end{aligned} $$
(f)
$$ \begin{aligned} & k^2-8 k+16 \\ & =(k-4)(k-4) \\ & =(k-4)^2 \end{aligned} $$
(g)
$$ \begin{aligned} & 2 m^2-11 m-6 \\ & =(2 m+1)(m-6) \end{aligned} $$
(h)
$$ \begin{aligned} & 9 f^2-12 f+4 \\ & =(3 f-2)(3 f-2) \\ & =(3 f-2)^2 \end{aligned} $$
(i)
$$ \begin{aligned} & 2 m^2+4 m-16 \\ & =(2 m-4)(m+4) \end{aligned} $$
(j)
$$ \begin{aligned} & 2 x^2-5 x-7 \\ & =(2 x-7)(x+1) \end{aligned} $$
(k)
$$ \begin{aligned} & 12 y^2+8 y-15 \\ & =(6 y-5)(2 y+3) \end{aligned} $$
(l)
$$ \begin{aligned} & 5 p^2+6 p-8 \\ & =(5 p-4)(p+2) \end{aligned} $$
(m)
$$ \begin{aligned} & -5 m^2-6 m+8 \\ & =(5 m-4)(-m-2) \end{aligned} $$
(n)
$$ \begin{aligned} & -3 p^2+8 p-4 \\ & =(3 p-2)(-p+2) \end{aligned} $$
(o)
$$ \begin{aligned} & -6 x^2-x+15 \\ & =(3 x+5)(-2 x+3) \end{aligned} $$
Faktorkan ungkapan algebra berikut.
(a) x2 + 9x + 14
(b) x2 + 7x – 18
(c) x2 – 5x – 24
(d) m2 + 11m – 26
(e) y2 – 2y – 15
(f) k2 – 8k + 16
(g) 2m2 – 11m – 6
(h) 9f2 – 12f + 4
(i) 2m2 + 4m – 16
(j) 2x2 – 5x – 7
(k) 12y2 + 8y – 15
(l) 5p2 + 6p – 8
(m) –5m2 – 6m + 8
(n) –3p2 + 8p − 4
(o) –6x2 – x + 15
Penyelesaian:
(a)
$$ \begin{aligned} & x^2+9 x+14 \\ & =(x+7)(x+2) \end{aligned} $$
(b)
$$ \begin{aligned} & x^2+7 x-18 \\ & =(x+9)(x-2) \end{aligned} $$
(c)
$$ \begin{aligned} & x^2-5 x-24 \\ & =(x+3)(x-8) \end{aligned} $$
(d)
$$ \begin{aligned} & m^2+11 m-26 \\ & =(m+13)(m-2) \end{aligned} $$
(e)
$$ \begin{aligned} & y^2-2 y-15 \\ & =(y+3)(y-5) \end{aligned} $$
(f)
$$ \begin{aligned} & k^2-8 k+16 \\ & =(k-4)(k-4) \\ & =(k-4)^2 \end{aligned} $$
(g)
$$ \begin{aligned} & 2 m^2-11 m-6 \\ & =(2 m+1)(m-6) \end{aligned} $$
(h)
$$ \begin{aligned} & 9 f^2-12 f+4 \\ & =(3 f-2)(3 f-2) \\ & =(3 f-2)^2 \end{aligned} $$
(i)
$$ \begin{aligned} & 2 m^2+4 m-16 \\ & =(2 m-4)(m+4) \end{aligned} $$
(j)
$$ \begin{aligned} & 2 x^2-5 x-7 \\ & =(2 x-7)(x+1) \end{aligned} $$
(k)
$$ \begin{aligned} & 12 y^2+8 y-15 \\ & =(6 y-5)(2 y+3) \end{aligned} $$
(l)
$$ \begin{aligned} & 5 p^2+6 p-8 \\ & =(5 p-4)(p+2) \end{aligned} $$
(m)
$$ \begin{aligned} & -5 m^2-6 m+8 \\ & =(5 m-4)(-m-2) \end{aligned} $$
(n)
$$ \begin{aligned} & -3 p^2+8 p-4 \\ & =(3 p-2)(-p+2) \end{aligned} $$
(o)
$$ \begin{aligned} & -6 x^2-x+15 \\ & =(3 x+5)(-2 x+3) \end{aligned} $$