จงแยกตัวประกอบของพหุนามต่อไปนี้
1) \(\small\mathsf{x^3 + 27}\)
วิธีทำ
\(\small\mathsf{x^3 + 27}\)
= \(\small\mathsf{x^3 + 3^3}\)
= \(\small\mathsf{(x + 3)(x^2 \, – \, 3x + 9)}\)
ตอบ \(\small\mathsf{(x + 3)(x^2 \, – \, 3x + 9)}\)
2) \(\small\mathsf{y^3 + 64}\)
วิธีทำ
\(\small\mathsf{y^3 + 64}\)
= \(\small\mathsf{y^3 + 4^3}\)
= \(\small\mathsf{(y + 4)(y^2 \, – \, 4y + 16)}\)
ตอบ \(\small\mathsf{(y + 4)(y^2 \, – \, 4y + 16)}\)
3) \(\small\mathsf{8x^3 + 1}\)
วิธีทำ
\(\small\mathsf{8x^3 + 1}\)
= \(\small\mathsf{{(2x)}^3 + 1^3}\)
= \(\small\mathsf{(2x + 1)(4x^2 \, – \, 2x + 1)}\)
ตอบ \(\small\mathsf{(2x + 1)(4x^2 \, – \, 2x + 1)}\)
4) \(\small\mathsf{64z^3 + 125}\)
วิธีทำ
\(\small\mathsf{64z^3 + 125}\)
= \(\small\mathsf{{(4z)}^3 + 5^3}\)
= \(\small\mathsf{(4z + 5)(16z^2 \, – \, 20z + 25)}\)
ตอบ \(\small\mathsf{(4z + 5)(16z^2 \, – \, 20z + 25)}\)
5) \(\small\mathsf{27x^3 + 512y^3}\)
วิธีทำ
\(\small\mathsf{27x^3 + 512y^3}\)
= \(\small\mathsf{{(3x)}^3 + {(8y)}^3}\)
= \(\small\mathsf{(3x + 8y)(9x^2 \, – \, 24xy + 64y^2)}\)
ตอบ \(\small\mathsf{(3x + 8y)(9x^2 \, – \, 24xy + 64y^2)}\)
6) \(\small\mathsf{729 + {(x \, – \, 2)}^3}\)
วิธีทำ
\(\small\mathsf{729 + {(x \, – \, 2)}^3}\)
= \(\small\mathsf{9^3 + {(x \, – \, 2)}^3}\)
= \(\small\mathsf{(9 + x \, – \, 2)[9^2 \, – \, 9(x \, – \, 2) + {(x \, – \, 2)}^2]}\)
= \(\small\mathsf{(7 + x)[81 \, – \, (9x \, – \, 18) + x^2 \, – \, 4x + 4]}\)
= \(\small\mathsf{(x + 7)(81 \, – \, 9x + 18 + x^2 \, – \, 4x + 4)}\)
= \(\small\mathsf{(x + 7)(x^2 \, – \, 13x + 103)}\)
ตอบ \(\small\mathsf{(x + 7)(x^2 \, – \, 13x + 103)}\)
7) \(\small\mathsf{{(3x \; – \, 1)}^3 + {(x \; – \, 4)}^3}\)
วิธีทำ
\(\small\mathsf{{(3x \; – \, 1)}^3 + {(x \; – \, 4)}^3}\)
= \(\small\mathsf{[(3x \; – \, 1) + (x \; – \, 4)][{(3x \; – \, 1)}^2 \; – \, (3x \; – \, 1)(x \; – \, 4) + {(x \; – \, 4)}^2]}\)
= \(\small\mathsf{(4x \; – \, 5)[9x^2 \; – \, 6x + 1 \; – \, (3x^2 \; – \, 12x \; – \, x + 4) + x^2 \; – \, 8x + 16]}\)
= \(\small\mathsf{(4x \; – \, 5)[9x^2 \; – \, 6x + 1 \; – \, (3x^2 \; – \, 13x + 4) + x^2 \; – \, 8x + 16]}\)
= \(\small\mathsf{(4x \; – \, 5)(9x^2 \; – \, 6x + 1 \; – \, 3x^2 + 13x \; – \, 4 + x^2 \; – \, 8x + 16)}\)
= \(\small\mathsf{(4x \; – \, 5)(7x^2 \; – \, x + 13)}\)
ตอบ \(\small\mathsf{(4x \; – \, 5)(7x^2 \; – \, x + 13)}\)
8) \(\small\mathsf{{(2x + 5)}^3 + {(5x \; – \, 9)}^3}\)
วิธีทำ
\(\small\mathsf{{(2x + 5)}^3 + {(5x \; – \, 9)}^3}\)
= \(\small\mathsf{[(2x + 5) + (5x \; – \, 9)][{(2x + 5)}^2 \; – \, (2x + 5)(5x \; – \, 9) + {(5x \; – \, 9)}^2]}\)
= \(\small\mathsf{(7x \; – \, 4)[4x^2 + 20x + 25 \; – \, (10x^2 \; – \, 18x + 25x \; – \, 45) + 25x^2 \; – \, 90x + 81]}\)
= \(\small\mathsf{(7x \; – \, 4)[4x^2 + 20x + 25 \; – \, (10x^2 + 7x \; – \, 45) + 25x^2 \; – \, 90x + 81]}\)
= \(\small\mathsf{(7x \; – \, 4)(4x^2 + 20x + 25 \; – \, 10x^2 \; – \, 7x + 45 + 25x^2 \; – \, 90x + 81)}\)
= \(\small\mathsf{(7x \; – \, 4)(19x^2 \; – \, 77x + 151)}\)
ตอบ \(\small\mathsf{(7x \; – \, 4)(19x^2 \; – \, 77x + 151)}\)
9) \(\small\mathsf{x^3 \; – \, 1}\)
วิธีทำ
\(\small\mathsf{x^3 \; – \, 1}\)
= \(\small\mathsf{x^3 \; – \, 1^3}\)
= \(\small\mathsf{(x \; – \, 1)(x^2 + x + 1)}\)
ตอบ \(\small\mathsf{(x \; – \, 1)(x^2 + x + 1)}\)
10) \(\small\mathsf{z^3 \; – \, 216}\)
วิธีทำ
\(\small\mathsf{z^3 \; – \, 216}\)
= \(\small\mathsf{z^3 \; – \, 6^3}\)
= \(\small\mathsf{(z \; – \, 6)(z^2 + 6z + 36)}\)
ตอบ \(\small\mathsf{(z \; – \, 6)(z^2 + 6z + 36)}\)
11) \(\small\mathsf{125y^3 + 64}\)
วิธีทำ
\(\small\mathsf{125y^3 + 64}\)
= \(\small\mathsf{{(5y)}^3 \; – \, 4^3}\)
= \(\small\mathsf{(5y \; – \, 4)(25y^2 + 20y + 16)}\)
ตอบ \(\small\mathsf{(5y \; – \, 4)(25y^2 + 20y + 16)}\)
12) 1,000 \(\small\mathsf{\, – \, 216x^3}\)
วิธีทำ
1,000 \(\small\mathsf{\, – \, 216x^3}\)
= \(\small\mathsf{{10}^3 \, – \, {(6x)}^3}\)
= \(\small\mathsf{(10 \, – \, 6x)(100 + 60x + 36x^2)}\)
= \(\small\mathsf{(2)(5 \, – \, 3x)(4)(25 + 15x + 9x^2)}\)
= \(\small\mathsf{8(5 \, – \, 3x)(9x^2 + 15x + 25)}\)
ตอบ \(\small\mathsf{8(5 \, – \, 3x)(9x^2 + 15x + 25)}\)
13) 1,331\(\small\mathsf{y^3 \; – \, 343z^3}\)
วิธีทำ
1,331\(\small\mathsf{y^3 \; – \, 343z^3}\)
= \(\small\mathsf{{(11y)}^3 \; – \, {(7z)}^3}\)
= \(\small\mathsf{(11y \; – \, 7z)(121y^2 + 77yz + 49z^2)}\)
ตอบ \(\small\mathsf{(11y \; – \, 7z)(121y^2 + 77yz + 49z^2)}\)
14) \(\small\mathsf{{(4x + 3)}^3 \; – \, 125}\)
วิธีทำ
\(\small\mathsf{{(4x + 3)}^3 \; – \, 125}\)
= \(\small\mathsf{{(4x + 3)}^3 \; – \, 5^3}\)
= \(\small\mathsf{(4x + 3 \; – \, 5)[{(4x + 3)}^2 + 5(4x + 3) + 25]}\)
= \(\small\mathsf{(4x \; – \, 2)(16x^2 + 24x + 9 + 20x + 15 + 25)}\)
= \(\small\mathsf{2(2x \; – \, 1)(16x^2 + 44x + 49)}\)
ตอบ \(\small\mathsf{2(2x \; – \, 1)(16x^2 + 44x + 49)}\)
15) \(\small\mathsf{8 \; – \, {(8x \; – \, 1)}^3}\)
วิธีทำ
\(\small\mathsf{8 \; – \, {(8x \; – \, 1)}^3}\)
= \(\small\mathsf{2^3 \; – \, {(8x \; – \, 1)}^3}\)
= \(\small\mathsf{[2 \; – \, (8x \; – \, 1)][4 + 2(8x \; – \, 1) + {(8x \; – \, 1)}^2]}\)
= \(\small\mathsf{(2 \; – \, 8x + 1)(4 + 16x \; – \, 2 + 64x^2 \; – \, 16x + 1)}\)
= \(\small\mathsf{(3 \; – \, 8x)(64x^2 + 3)}\)
ตอบ \(\small\mathsf{(3 \; – \, 8x)(64x^2 + 3)}\)
16) \(\small\mathsf{{(8x + 1)}^3 + 8}\)
วิธีทำ
\(\small\mathsf{{(8x + 1)}^3 + 8}\)
= \(\small\mathsf{{(8x + 1)}^3 + 2^3}\)
= \(\small\mathsf{(8x + 1 + 2)[{(8x + 1)}^2 \; – \, 2(8x + 1) + 4]}\)
= \(\small\mathsf{(8x + 3)[64x^2 + 16x + 1 \; – \, (16x + 2) + 4]}\)
= \(\small\mathsf{(8x + 3)(64x^2 + 16x + 1 \; – \, 16x \; – \, 2 + 4)}\)
= \(\small\mathsf{(8x + 3)(64x^2 + 3)}\)
ตอบ \(\small\mathsf{(8x + 3)(64x^2 + 3)}\)
17) \(\small\mathsf{{(5x \; – \, 2)}^3 + {(5x + 2)}^3}\)
วิธีทำ
\(\small\mathsf{{(5x \; – \, 2)}^3 + {(5x + 2)}^3}\)
= (5x – 2 + 5x + 2)\(\small\mathsf{[{(5x \; – \, 2)}^2 \; – \, (5x \; – \, 2)(5x + 2) + (5x + 2)^2]}\)
= \(\small\mathsf{10x[25x^2 \; – \, 20x + 4 \; – \, (25x^2 \; – \, 4) + 25x^2 + 20x + 4]}\)
= \(\small\mathsf{10x(25x^2 \; – \, 20x + 4 \; – \, 25x^2 + 4 + 25x^2 + 20x + 4)}\)
= \(\small\mathsf{10x(25x^2 + 12)}\)
ตอบ \(\small\mathsf{10x(25x^2 + 12)}\)
18) \(\small\mathsf{{(2x + 5)}^3 \; – \, {(2x \; – \, 5)}^3}\)
วิธีทำ
\(\small\mathsf{{(2x + 5)}^3 \; – \, {(2x \; – \, 5)}^3}\)
= [(2x + 5) – (2x – 5)]\(\small\mathsf{[{(2x + 5)}^2 + (2x + 5)(2x \; – \, 5) + {(2x \; – \, 5)}^2]}\)
= (2x + 5 – 2x + 5)\(\small\mathsf{[4x^2 + 20x + 25 + (4x^2 \; – \, 25) + 4x^2 \; – \, 20x + 25]}\)
= \(\small\mathsf{10(12x^2 + 25)}\)
ตอบ \(\small\mathsf{10x(25x^2 + 12)}\)
19) \(\small\mathsf{{(7x \; – \, 2)}^3 \; – \, {(6x + 9)}^3}\)
วิธีทำ
\(\small\mathsf{{(7x \; – \, 2)}^3 \; – \, {(6x + 9)}^3}\)
= \(\small\mathsf{[(7x \; – \, 2) \; – \, (6x + 9)][{(7x \; – \, 2)}^2 + (7x \; – \, 2)(6x + 9) + {(6x + 9)}^2]}\)
= \(\small\mathsf{(7x – 2 – 6x – 9)(49x^2 \; – \, 28x + 4 + 42x^2 + 63x – 12x – 18 + 36x^2 + 108x + 81)}\)
= \(\small\mathsf{(x \; – \, 11)(127x^2 + 131x + 67)}\)
ตอบ \(\small\mathsf{(x \; – \, 11)(127x^2 + 131x + 67)}\)
20) \(\small\mathsf{{(8x \; – \, 15)}^3 \; – \, {(3x \; – \, 7)}^3}\)
วิธีทำ
\(\small\mathsf{{(8x – 15)}^3 – {(3x – 7)}^3}\)
= \(\small\mathsf{[(8x \; – \, 15) \; – \, (3x \; – \, 7)][{(8x \; – \, 15)}^2 + (8x \; – \, 15)(3x \; – \, 7) + {(3x \; – \, 7)}^2]}\)
= \(\small\mathsf{(8x – 15 – 3x + 7)(64x^2 – 240x + 225 + 24x^2 – 56x – 45x + 105 + 9x^2 – 42x + 49)}\)
= \(\small\mathsf{(5x \; – \, 8)(97x^2 \; – \, 383x + 379)}\)
ตอบ \(\small\mathsf{(5x \; – \, 8)(97x^2 \; – \, 383x + 379)}\)
21) \(\small\mathsf{x^3 \; – \, x^2 \; – \, x + 1}\)
วิธีทำ
\(\small\mathsf{x^3 \; – \, x^2 \; – \, x + 1}\)
= \(\small\mathsf{(x^3 + 1) \; – \, (x^2 + x)}\)
= \(\small\mathsf{[(x + 1)(x^2 \; – \, x + 1)] \; – \, x(x + 1)}\)
= \(\small\mathsf{(x + 1)(x^2 \; – \, x + 1 \; – \, x)}\)
= \(\small\mathsf{(x + 1)(x^2 \; – \, 2x + 1)}\)
= (x + 1)(x – 1)(x – 1)
= \(\small\mathsf{(x + 1){(x \; – \, 1)}^2}\)
ตอบ \(\small\mathsf{(x + 1){(x \; – \, 1)}^2}\)
22) \(\small\mathsf{y^3 + y^2 \; – \, 4y \; – \, 64}\)
วิธีทำ
\(\small\mathsf{y^3 + y^2 \; – \, 4y \; – \, 64}\)
= \(\small\mathsf{(y^3 \; – \, 64) + (y^2 \; – \, 4y)}\)
= \(\small\mathsf{(y^3 \; – \, 4^3) + y(y \; – \, 4)}\)
= \(\small\mathsf{(y \; – \, 4)(y^2 + 4y + 16) + y(y \; – \, 4)}\)
= \(\small\mathsf{(y \; – \, 4)(y^2 + 4y + 16 + y)}\)
= \(\small\mathsf{(y \; – \, 4)(y^2 + 5y + 16)}\)
ตอบ \(\small\mathsf{(y \; – \, 4)(y^2 + 5y + 16)}\)