จงแยกตัวประกอบของพหุนามต่อไปนี้
1) \(\small\mathsf{x^4 + 3x^2}\)
วิธีทำ
\(\small\mathsf{x^4 + 3x^2}\)
= \(\small\mathsf{x^2(x^2 + 3)}\)
ตอบ \(\small\mathsf{x^2(x^2 + 3)}\)
2) \(\small\mathsf{4x^4 \; – \, 16x^2}\)
วิธีทำ
\(\small\mathsf{4x^4 \; – \, 16x^2}\)
= \(\small\mathsf{4x^2(x^2 \; – \, 4)}\)
= \(\small\mathsf{4x^2(x + 2)(x \; – \, 2)}\)
ตอบ \(\small\mathsf{4x^2(x + 2)(x \; – \, 2)}\)
3) \(\small\mathsf{x^6 \; – \, 81x^4}\)
วิธีทำ
\(\small\mathsf{x^6 \; – \, 81x^4}\)
= \(\small\mathsf{x^4(x^2 \; – \, 81)}\)
= \(\small\mathsf{x^4(x + 9)(x \; – \, 9)}\)
ตอบ \(\small\mathsf{x^4(x + 9)(x \; – \, 9)}\)
4) \(\small\mathsf{x^6 + 2x^4 + x^2}\)
วิธีทำ
\(\small\mathsf{x^6 + 2x^4 + x^2}\)
= \(\small\mathsf{x^2(x^4 + 2x^2 + 1)}\)
= \(\small\mathsf{x^2(x^2 + 1)(x^2 + 1)}\)
= \(\small\mathsf{x^2{(x^2 + 1)}^2}\)
ตอบ \(\small\mathsf{x^2{(x^2 + 1)}^2}\)
5) \(\small\mathsf{x^4 \; – \, 625}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, 625}\)
= \(\small\mathsf{{(x^2)}^2 \; – \, {25}^2}\)
= \(\small\mathsf{(x^2 + 25)(x^2 \; – \, 25)}\)
= \(\small\mathsf{(x^2 + 25)(x + 5)(x \; – \, 5)}\)
ตอบ \(\small\mathsf{(x^2 + 25)(x + 5)(x \; – \, 5)}\)
6) \(\small\mathsf{81y^4 \; – \, 625}\)
วิธีทำ
\(\small\mathsf{81y^4 \; – \, 625}\)
= \(\small\mathsf{{(9y^2)}^2 \; – \, {25}^2}\)
= \(\small\mathsf{(9y^2 + 25)(9y^2 \; – \, 25)}\)
= \(\small\mathsf{(9y^2 + 25)[{(3y)}^2 \; – \, 5^2]}\)
= \(\small\mathsf{(9y^2 + 25)(3y + 5)(3y \; – \, 5)}\)
ตอบ \(\small\mathsf{(9y^2 + 25)(3y + 5)(3y \; – \, 5)}\)
7) \(\small\mathsf{160,000x^4 \; – \, 10,000}\)
วิธีทำ
\(\small\mathsf{160,000x^4 \; – \, 10,000}\)
= \(\small\mathsf{10,000(16x^4 \; – \, 1)}\)
= \(\small\mathsf{10,000[{(4x^2)}^2 \; – \, 1^2]}\)
= \(\small\mathsf{10,000(4x^2 + 1)(4x^2 \; – \, 1)}\)
= \(\small\mathsf{10,000(4x^2 + 1)[{(2x)}^2 \; – \, 1^2]}\)
= \(\small\mathsf{10,000(4x^2 + 1)(2x + 1)(2x \; – \, 1)}\)
ตอบ \(\small\mathsf{10,000(4x^2 + 1)(2x + 1)(2x \; – \, 1)}\)
8) \(\small\mathsf{x^4 \; – \, 50x^2 + 625}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, 50x^2 + 625}\)
= \(\small\mathsf{{(x^2)}^2 \; – \, 2(x^2)(25) + {25}^2}\)
= \(\small\mathsf{{(x^2 \; – \, 25)}^2}\)
= \(\small\mathsf{{[(x + 5)(x \; – \, 5)]}^2}\)
= \(\small\mathsf{{(x + 5)}^2{(x \; – \, 5)}^2}\)
ตอบ \(\small\mathsf{{(x + 5)}^2{(x \; – \, 5)}^2}\)
9) \(\small\mathsf{25y^4 + 120y^2 + 144}\)
วิธีทำ
\(\small\mathsf{25y^4 + 120y^2 + 144}\)
= \(\small\mathsf{{(5y^2)}^2 + 2(5y^2)(12) + {12}^2}\)
= \(\small\mathsf{{(5y^2 + 12)}^2}\)
ตอบ \(\small\mathsf{{(5y^2 + 12)}^2}\)
10) \(\small\mathsf{77x^4 + 170x^2 + 77}\)
วิธีทำ
\(\small\mathsf{77x^4 + 170x^2 + 77}\)
= \(\small\mathsf{(7x^2 + 11)(11x^2 + 7)}\)
ตอบ \(\small\mathsf{(7x^2 + 11)(11x^2 + 7)}\)
11) \(\small\mathsf{77x^4 \; – \, 72x^2 \; – \, 77}\)
วิธีทำ
\(\small\mathsf{77x^4 \; – \, 72x^2 \; – \, 77}\)
= \(\small\mathsf{(7x^2 \; – \, 11)(11x^2 + 7)}\)
ตอบ \(\small\mathsf{(7x^2 \; – \, 11)(11x^2 + 7)}\)
12) \(\small\mathsf{x^4 \; – \, 10x^2 + 9}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, 10x^2 + 9}\)
= \(\small\mathsf{(x^2 \; – \, 9)(x^2 \; – \, 1)}\)
= (x + 3)(x – 3)(x + 1)(x – 1)
ตอบ (x + 3)(x – 3)(x + 1)(x – 1)
13) \(\small\mathsf{x^4 \; – \, 29x^2 + 100}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, 29x^2 + 100}\)
= \(\small\mathsf{(x^2 \; – \, 25)(x^2 \; – \, 4)}\)
= (x + 5)(x – 5)(x + 2)(x – 2)
ตอบ (x + 5)(x – 5)(x + 2)(x – 2)
14) \(\small\mathsf{64x^6 \; – \, 729}\)
วิธีทำ
\(\small\mathsf{64x^6 \; – \, 729}\)
= \(\small\mathsf{{(8x^3)}^2 \; – \, {27}^2}\)
= \(\small\mathsf{(8x^3 + 27)(8x^3 \; – \, 27)}\)
= \(\small\mathsf{[{(2x)}^3 + 3^3][{(2x)}^3 \; – \, 3^3]}\)
= \(\small\mathsf{(2x + 3)(4x^2 \; – \, 6x + 9)(2x \; – \, 3)(4x^2 + 6x + 9)}\)
= \(\small\mathsf{(2x + 3)(2x \; – \, 3)(4x^2 \; – \, 6x + 9)(4x^2 + 6x + 9)}\)
ตอบ \(\small\mathsf{(2x + 3)(2x \; – \, 3)(4x^2 \; – \, 6x + 9)(4x^2 + 6x + 9)}\)
15) \(\small\mathsf{x^6 + 216}\)
วิธีทำ
\(\small\mathsf{x^6 + 216}\)
= \(\small\mathsf{{(x^2)}^3 + 6^3}\)
= \(\small\mathsf{(x^2 + 6)(x^4 \; – \, 6x^2 + 36)}\)
ตอบ \(\small\mathsf{(x^2 + 6)(x^4 \; – \, 6x^2 + 36)}\)
16) \(\small\mathsf{512 \; – \, y^6}\)
วิธีทำ
\(\small\mathsf{512 \; – \, y^6}\)
= \(\small\mathsf{2^9 \; – \, y^6}\)
= \(\small\mathsf{{(2^3)}^3 \; – \, {(y^2)}^3}\)
= \(\small\mathsf{(2^3 \; – \, y^2)(2^6 + 2^3y^2 + y^4)}\)
= \(\small\mathsf{(8 \; – \, y^2)(y^4 + 8y^2 + 64)}\)
ตอบ \(\small\mathsf{(8 \; – \, y^2)(y^4 + 8y^2 + 64)}\)
17) \(\small\mathsf{x^4 \; – \, y^4}\)
วิธีทำ
\(\small\mathsf{x^4 \; – \, y^4}\)
= \(\small\mathsf{{(x^2)}^2 \; – \, {(y^2)}^2}\)
= \(\small\mathsf{(x^2 + y^2)(x^2 \; – \, y^2)}\)
= \(\small\mathsf{(x^2 + y^2)(x + y)(x \; – \, y)}\)
ตอบ \(\small\mathsf{(x^2 + y^2)(x + y)(x \; – \, y)}\)
18) \(\small\mathsf{a^6b^3 \; – \, 1}\)
วิธีทำ
\(\small\mathsf{a^6b^3 \; – \, 1}\)
= \(\small\mathsf{{(a^2b)}^3 \; – \, 1^3}\)
= \(\small\mathsf{(a^2b \; – \, 1)(a^4b^2 + a^2b + 1)}\)
ตอบ \(\small\mathsf{(a^2b \; – \, 1)(a^4b^2 + a^2b + 1)}\)
19) \(\small\mathsf{x^6 \; – \, y^6}\)
วิธีทำ
\(\small\mathsf{x^6 \; – \, y^6}\)
= \(\small\mathsf{{(x^3)}^2 \; – \, {(y^3)}^2}\)
= \(\small\mathsf{(x^3 + y^3)(x^3 \; – \, y^3)}\)
= \(\small\mathsf{(x + y)(x^2 \; – \, xy + y^2)(x \; – \, y)(x^2 + xy + y^2)}\)
= \(\small\mathsf{(x + y)(x \; – \, y)(x^2 \; – \, xy + y^2)(x^2 + xy + y^2)}\)
ตอบ \(\small\mathsf{(x + y)(x \; – \, y)(x^2 \; – \, xy + y^2)(x^2 + xy + y^2)}\)
20) \(\small\mathsf{216x^6 \; – \, 27y^6}\)
วิธีทำ
\(\small\mathsf{216x^6 \; – \, 27y^6}\)
= \(\small\mathsf{27(8x^6 \; – \, y^6)}\)
= \(\small\mathsf{27[{(2x^2)}^3 \; – \, {(y^2)}^3]}\)
= \(\small\mathsf{27(2x^2 \; – \, y^2)(4x^4 + 2x^2y^2 + y^4)}\)
ตอบ \(\small\mathsf{27(2x^2 \; – \, y^2)(4x^4 + 2x^2y^2 + y^4)}\)