1. จงแยกตัวประกอบของพหุนามต่อไปนี้
1) \(\small\mathsf{8 \; – \, x^3}\)
วิธีทำ
\(\small\mathsf{8 \; – \, x^3}\)
= \(\small\mathsf{2^3 \; – \, x^3}\)
= \(\small\mathsf{(2 \; – \, x)(4 + 2x + x^2)}\)
= \(\small\mathsf{(2 \; – \, x)(x^2 + 2x + 4)}\)
ตอบ \(\small\mathsf{(2 \; – \, x)(x^2 + 2x + 4)}\)
2) \(\small\mathsf{27x^3 + 125}\)
วิธีทำ
\(\small\mathsf{27x^3 + 125}\)
= \(\small\mathsf{{(3x)}^3 + 5^3}\)
= \(\small\mathsf{(3x + 5)(9x^2 \; – \, 15x + 25)}\)
ตอบ \(\small\mathsf{(3x + 5)(9x^2 \; – \, 15x + 25)}\)
3) \(\small\mathsf{{(x \; – \, 3)}^3 + 27}\)
วิธีทำ
\(\small\mathsf{{(x \; – \, 3)}^3 + 27}\)
= \(\small\mathsf{{(x \; – \, 3)}^3 + 3^3}\)
= \(\small\mathsf{(x \; – \, 3 + 3)[{(x \; – \, 3)}^2 \; – \, 3(x \; – \, 3) + 9]}\)
= \(\small\mathsf{x(x^2 \; – \, 6x + 9 \; – \, 3x + 9 + 9)}\)
= \(\small\mathsf{x(x^2 \; – \, 9x + 27)}\)
ตอบ \(\small\mathsf{x(x^2 \; – \, 9x + 27)}\)
4) 1,000 \(\small\mathsf{\; – \, {(x + 10)}^3}\)
วิธีทำ
1,000 \(\small\mathsf{\; – \, {(x + 10)}^3}\)
= \(\small\mathsf{{10}^3 \; – \, {(x + 10)}^3}\)
= \(\small\mathsf{[10 \; – \, (x + 10)][100 + 10(x + 10) + {(x + 10)}^2]}\)
= \(\small\mathsf{(10 \; – \, x \; – \, 10)(100 + 10x + 100 + x^2 + 20x + 100)}\)
= \(\small\mathsf{-x(x^2 + 30x + 300)}\)
ตอบ \(\small\mathsf{-x(x^2 + 30x + 300)}\)
5) \(\small\mathsf{{(x \; – \, 11)}^3 \; – \, {(x + 11)}^3}\)
วิธีทำ
\(\small\mathsf{{(x \; – \, 11)}^3 \; – \, {(x + 11)}^3}\)
= \(\small\mathsf{[(x \; – \, 11) \; – \, (x + 11)][{(x \; – \, 11)}^2 + (x \; – \, 11)(x + 11) + {(x + 11)}^2]}\)
= \(\small\mathsf{(x \; – \, 11 \; – \, x \; – \, 11)[x^2 \; – \, 22x + 121 + (x^2 \; – \, 121) + x^2 + 22x + 121]}\)
= \(\small\mathsf{-22(3x^2 + 121)}\)
ตอบ \(\small\mathsf{-22(3x^2 + 121)}\)
6) \(\small\mathsf{{(x \; – \, 5)}^3 \; – \, {(7x + 4)}^3}\)
วิธีทำ
\(\small\mathsf{{(x \; – \, 5)}^3 \; – \, {(7x + 4)}^3}\)
= \(\small\mathsf{[(x \; – \, 5) \; – \, (7x + 4)][{(x \; – \, 5)}^2 + (x \; – \, 5)(7x + 4) + {(7x + 4)}^2]}\)
= \(\small\mathsf{(x \; – \, 5 \; – \, 7x \; – \, 4)(x^2 \; – \, 10x + 25 + 7x^2 + 4x \; – \, 35x \; – \, 20 + 49x^2 + 56x + 16)}\)
= \(\small\mathsf{(-6x \; – \, 9)(57x^2 + 15x + 21)}\)
= \(\small\mathsf{(-3)(2x + 3)(3)(19x^2 + 5x + 7)}\)
= \(\small\mathsf{-9(2x + 3)(19x^2 + 5x + 7)}\)
ตอบ \(\small\mathsf{-9(2x + 3)(19x^2 + 5x + 7)}\)
7) \(\small\mathsf{81x^4 \; – \,}\) 2,401
วิธีทำ
\(\small\mathsf{81x^4 \; – \,}\) 2,401
= \(\small\mathsf{{(9x^2)}^2 \; – \, {49}^2}\)
= \(\small\mathsf{(9x^2 + 49)(9x^2 \; – \, 49)}\)
= \(\small\mathsf{(9x^2 + 49)[{(3x)}^2 \; – \, 7^2]}\)
= \(\small\mathsf{(9x^2 + 49)(3x + 7)(3x \; – \, 7)}\)
ตอบ \(\small\mathsf{(9x^2 + 49)(3x + 7)(3x \; – \, 7)}\)
8) \(\small\mathsf{x^4 \; – \, 17x^2 + 16}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, 17x^2 + 16}\)
= \(\small\mathsf{(x^2 \; – \, 16)(x^2 \; – \, 1)}\)
= (x + 4)(x – 4)(x + 1)(x – 1)
ตอบ (x + 4)(x – 4)(x + 1)(x – 1)
9) \(\small\mathsf{x^4 \; – \, 32x^2 + 256}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, 32x^2 + 256}\)
= \(\small\mathsf{(x^2 \; – \, 16)(x^2 \; – \, 16)}\)
= (x + 4)(x – 4)(x + 4)(x – 4)
= \(\small\mathsf{{(x + 4)}^2{(x \; – \, 4)}^2}\)
ตอบ \(\small\mathsf{{(x + 4)}^2{(x \; – \, 4)}^2}\)
10) \(\small\mathsf{10x^4 \; – \, x^2 \; – \, 3}\)
วิธีทำ
\(\small\mathsf{10x^4 \; – \, x^2 \; – \, 3}\)
= \(\small\mathsf{(2x^2 + 1)(5x^2 \; – \, 3)}\)
ตอบ \(\small\mathsf{(2x^2 + 1)(5x^2 \; – \, 3)}\)
11) \(\small\mathsf{36x^4 \; – \, 97x^2 + 36}\)
วิธีทำ
\(\small\mathsf{36x^4 \; – \, 97x^2 + 36}\)
= \(\small\mathsf{(9x^2 \; – \, 4)(4x^2 \; – \, 9)}\)
= \(\small\mathsf{[{(3x)}^2 \; – \, 2^2][{(2x)}^2 \; – \, 3^2]}\)
= (3x + 2)(3x – 2)(2x + 3)(2x – 3)
ตอบ (3x + 2)(3x – 2)(2x + 3)(2x – 3)
12) \(\small\mathsf{y^6 \; – \,}\) 4,096
วิธีทำ
\(\small\mathsf{y^6 \; – \,}\) 4,096
= \(\small\mathsf{y^6 \; – \, 4^6}\)
= \(\small\mathsf{{(y^3)}^2 \; – \, {(4^3)}^2}\)
= \(\small\mathsf{(y^3 + 4^3)(y^3 \; – \, 4^3)}\)
= \(\small\mathsf{(y + 4)(y^2 \; – \, 4y + 16)(y \; – \, 4)(y^2 + 4y + 16)}\)
= \(\small\mathsf{(y + 4)(y \; – \, 4)(y^2 \; – \, 4y + 16)(y^2 + 4y + 16)}\)
ตอบ \(\small\mathsf{(y + 4)(y \; – \, 4)(y^2 \; – \, 4y + 16)(y^2 + 4y + 16)}\)
13) \(\small\mathsf{1 \; – \, a^6}\)
วิธีทำ
\(\small\mathsf{1 \; – \, a^6}\)
= \(\small\mathsf{1^2 \; – \, {(a^3)}^2}\)
= \(\small\mathsf{(1 + a^3)(1 \; – \, a^3)}\)
= \(\small\mathsf{(1^3 + a^3)(1^3 \; – \, a^3)}\)
= \(\small\mathsf{(1 + a)(1 \; – \, a + a^2)(1 \; – \, a)(1 + a + a^2)}\)
= \(\small\mathsf{(a + 1)(1 \; – \, a)(a^2 \; – \, a + 1)(a^2 + a + 1)}\)
ตอบ \(\small\mathsf{(a + 1)(1 \; – \, a)(a^2 \; – \, a + 1)(a^2 + a + 1)}\)
14) 1,000,000\(\small\mathsf{z^6 \; – \, 1}\)
วิธีทำ
1,000,000\(\small\mathsf{z^6 \; – \, 1}\)
= \(\small\mathsf{{(10z)}^6 \; – \, 1}\)
= \(\small\mathsf{{({(10z)}^3)}^2 \; – \, 1^2}\)
= \(\small\mathsf{[{(10z)}^3 + 1][{(10z)}^3 \; – \, 1]}\)
= \(\small\mathsf{[{(10z)}^3 + 1^3][{(10z)}^3 \; – \, 1^3]}\)
= \(\small\mathsf{(10z + 1)(100z^2 \; – \, 10z + 1)(10z \; – \, 1)(100z^2 + 10z + 1)}\)
= \(\small\mathsf{(10z + 1)(10z \; – \, 1)(100z^2 \; – \, 10z + 1)(100z^2 + 10z + 1)}\)
ตอบ \(\small\mathsf{(10z + 1)(10z \; – \, 1)(100z^2 \; – \, 10z + 1)(100z^2 + 10z + 1)}\)
15) \(\small\mathsf{x^6 + 6x^4 + 9x^2}\)
วิธีทำ
\(\small\mathsf{x^6 + 6x^4 + 9x^2}\)
= \(\small\mathsf{x^2(x^4 + 6x^2 + 9)}\)
= \(\small\mathsf{x^2(x^2 + 3)(x^2 + 3)}\)
= \(\small\mathsf{x^2{(x^2 + 3)}^2}\)
ตอบ \(\small\mathsf{x^2{(x^2 + 3)}^2}\)
16) \(\small\mathsf{25x^6 + 10x^3 + 1}\)
วิธีทำ
\(\small\mathsf{25x^6 + 10x^3 + 1}\)
= \(\small\mathsf{(5x^3 + 1)(5x^3 + 1)}\)
= \(\small\mathsf{{(5x^3 + 1)}^2}\)
ตอบ \(\small\mathsf{{(5x^3 + 1)}^2}\)
17) \(\small\mathsf{81x^4 \; – \, 256y^4}\)
วิธีทำ
\(\small\mathsf{81x^4 \; – \, 256y^4}\)
= \(\small\mathsf{{(9x^2)}^2 \; – \, {(16y^2)}^2}\)
= \(\small\mathsf{(9x^2 + 16y^2)(9x^2 \; – \, 16y^2)}\)
= \(\small\mathsf{(9x^2 + 16y^2)[{(3x)}^2 \; – \, {(4y)}^2]}\)
= \(\small\mathsf{(9x^2 + 16y^2)(3x + 4y)(3x \; – \, 4y)}\)
ตอบ \(\small\mathsf{(9x^2 + 16y^2)(3x + 4y)(3x \; – \, 4y)}\)
18) \(\small\mathsf{64 \; – \, x^3y^3}\)
วิธีทำ
\(\small\mathsf{64 \; – \, x^3y^3}\)
= \(\small\mathsf{4^3 \; – \, {(xy)}^3}\)
= \(\small\mathsf{(4 \; – \, xy)(16 + 4xy + x^2y^2)}\)
= \(\small\mathsf{(4 \; – \, xy)(x^2y^2 + 4xy + 16)}\)
ตอบ \(\small\mathsf{(4 \; – \, xy)(x^2y^2 + 4xy + 16)}\)
19) \(\small\mathsf{343x^6 +}\) 1,000\(\small\mathsf{z^6}\)
วิธีทำ
\(\small\mathsf{343x^6 +}\) 1,000\(\small\mathsf{z^6}\)
= \(\small\mathsf{{(7x^2)}^3 + {(10z^2)}^3}\)
= \(\small\mathsf{(7x^2 + 10z^2)(49x^4 \; – \, 70x^2z^2 + 100z^4)}\)
ตอบ \(\small\mathsf{(7x^2 + 10z^2)(49x^4 \; – \, 70x^2z^2 + 100z^4)}\)
20) \(\small\mathsf{x^6 \; – \, 4x^3y + 4y^2}\)
วิธีทำ
\(\small\mathsf{x^6 \; – \, 4x^3y + 4y^2}\)
= \(\small\mathsf{{(x^3)}^2 \; – \, 2(x^3)(2y) + {(2y)}^2}\)
= \(\small\mathsf{{(x^3 \; – \, 2y)}^2}\)
ตอบ \(\small\mathsf{{(x^3 \; – \, 2y)}^2}\)