Question

把下列各式分解因式①a(a-b)2-b(b-a)2②a(a＋b）（a－b)－b(b＋a）（b－a）

③x²y（x－y）－x²y²（x－y）＋xy²（y－x）④﹣a²（x²＋y²）ab（y²＋x²）－ac（x²＋y²)⑤m（b＋c－d）－n（d－b－c）－b－c＋d⑥3x的n次方乘（1－x）－2(x的n次方减x的n＋1次方）⑥（a－2b＋c）²－(a＋2b－c）² ⑺（a²－b²）（a＋b）－(a－b）的三次方⑧（16a²－9b²）＋（4a＋3b)⑨（4a²－9b²）（3b－2a）＋9（2a＋3b）²⑽（3a－1）²－（12a－4）；11、（a－b）的2n＋2次方减（b－a）的2n次方；12、9x²＋16（x＋y)²－24x（x＋y）；13、x的2n次方减2x的n次方y的2n次方加y的4n次方；14、4x²y²－（x²－y²）²；15﹣2ax的n－1次方减18ax的n＋1次方＋12ax的n次方

Answer 1

a(a - b)" - b(b - a)"
= a(a - b)" - b(a - b)"
= (a - b)^3

a(a + b)(a - b) - b(b + a)(b - a)
= a(a + b)(a - b) + b(a + b)(a - b)
= (a + b)"(a - b)

x"y(x - y) - x"y"(x - y) + xy"(y - x)
= x"y(x - y) - x"y"(x - y) - xy"(x - y)
= xy(x - y)(x - xy - y)

-a"(x" + y") - ab(y" + x") - ac(x" + y")
= -a(x" + y")(a + b + c)

m(b + c - d) - n(d - b - c) - b - c + d
= m(b + c - d) + n(b + c - d) - (b + c - d)
= (b + c - d)(m + n - 1)

3(x^n)(1 - x) - 2[ x^n - x^(n+1) ]
= 3(x^n)(1 - x) - 2(x^n)(1 - x)
= (x^n)(1 - x )

(a - 2b + c)" - (a + 2b - c)"
= (a - 2b + c + a + 2b - c)(a - 2b + c - a - 2b + c)
= 2a( 2c - 4b )
= 4a( c - 2b )

(a" - b")(a + b) - (a - b)^3
= (a + b)"(a - b) - (a - b)^3
= (a - b)[ (a + b)" - (a - b)" ]
= (a - b)(a + b + a - b)(a + b - a + b)
= (a - b)(2a)(2b)
= 4ab(a - b)

(16a" - 9b") + (4a + 3b)
= (4a + 3b)(4a - 3b) + (4a + 3b)
= (4a + 3b)(4a - 3b + 1)

(4a" - 9b")(3b - 2a) + 9(2a + 3b)"
= (2a + 3b)(2a - 3b)(3b - 2a) + 9(2a + 3b)"
= (2a + 3b)[ -(2a - 3b)" + 9(2a + 3b) ]
= (2a + 3b)[ 9(2a + 3b) - (2a - 3b)" ]

(3a - 1)" - (12a - 4)
= (3a - 1)" - 4(3a - 1)
= (3a - 1)(3a - 1 - 4)
= (3a - 1)(3a - 5)

(a - b)^(2n+2) - (b - a)^(2n)
= (a - b)"(a - b)^(2n) - (a - b)^(2n)
= (a - b)^(2n)[ (a - b)" - 1 ]
= (a - b + 1)(a - b - 1)(a - b)^(2n)

9x" + 16(x + y)" - 24x(x + y)
= (3x)" + [4(x + y)]" - 2(3x)(4x + 4y)
= [ 3x - (4x + 4y) ]"
= (3x - 4x + 4y)"
= (-x + 4y)"
= (x - 4y)"

(x^2n) - 2(x^n)(y^2n) + (y^4n)
= (x^n)" - 2(x^n)(y^2n) + (y^2n)"
= (x^n - y^2n)"

4x"y" - (x" - y")"
= (2xy)" - (x" - y")"
= (2xy + x" - y")(2xy - x" + y")

-2ax^(n-1) - 18ax^(n+1) + 12ax^n
= -2a[ x^(n-1) ] - 18ax"[ x^(n-1) ] + 12ax[ x^(n-1) ]
= -2a[ x^(n-1) ]( 1 - 9x" + 6x )
= -2a[ x^(n-1) ](3x - 1)"

Answer 2

①a(a-b)²-b(b-a)²
=a(a-b)²-b(a-b)²
=(a-b)(a-b)²
=(a-b)³

②a(a+b)(a-b)-b(b+a)(b-a)
=a(a+b)(a-b)+b(a+b)(a-b)
=(a+b)(a+b)(a-b)
=(a+b)²(a-b)

③x²y(x-y)-x²y²(x-y)+xy²(y-x)
=x²y(x-y)-x²y²(x-y)-xy²(x-y)
=(x-y)(x²y-x²y²-xy²)
=xy(x-y)(x-xy-y)

④a²(x²+y²)-ab(y²+x²)-ac(x²+y²) （题目对否？）
=a(a-b-c)(x²+y²)

⑤m(b+c-d)-n(d-b-c)-b-c+d
=m(b+c-d)-n[d-(b+c)]-(b+c-d)
=m(b+c-d)+n(b+c-d)-(b+c-d)
=(b+c-d)(m+n-1)

⑥(a-2b+c)²-(a+2b-c)²
=[(a-2b+c)+(a+2b-c)][(a-2b+c)-(a+2b-c)]
=2a(-4b+2c)
=4a(c-2b)

⑺(a²-b²)(a+b)-(a-b)³
=(a+b)(a-b)(a+b)-(a-b)³
=(a+b)²(a-b)-(a-b)³
=(a-b)[(a+b)²-(a-b)²]
=(a-b)[(a+b)+(a-b)][(a+b)-(a-b)]
=4ab(a-b)

⑧(16a²-9b²)+(4a+3b)
=(4a+3b)(4a-3b)+(4a+3b)
=(4a+3b)(4a-3b+1)

⑨(4a²-9b²)(3b-2a)+9(2a+3b)²
=-(2a+3b)(2a-3b)(2a+3b)+9(2a+3b)²
=(2a+3b)²[9-(2a-3b)]
=(2a+3b)²(9-2a+3b)

⑽(3a-1)²-(12a-4)
=(3a-1)²-4(3a-1)
=(3a-1)(3a-1-4)
=(3a-1)(3a-5)

追问

⑨（4a²－9b²）（3b－2a）＋9（2a＋3b）²⑽（3a－1）²－（12a－4）谢谢

追答

11、（a－b）的（2n＋2）次方－（b－a）的2n次方
=（a－b）的（2n＋2）次方－（a－b）的2n次方
=（a－b）的2n次方[(a-b)²-1]

12、9x²＋16（x＋y)²－24x（x＋y）
=9x²+16x²+32xy+16y²-24x²-24xy
=x²+8xy+16y²
=(x+4y)²

13、(x的2n次方-2x的n次方)(y的2n次方+y的4n次方)
=x的2n次方(x的2n次方-2)(y的2n次方)(1+y²)

14、4x²y²+(x²-y²)² （题目对否？我修改了）
=4ab+(a-b) 用a代替x²，用b代替y²
=4ab+(a² -2ab+b² )
=a² +2ab+b²
=(a+b)²
=(x²+y²)²

15、﹣2ax的n-1次方-18ax的n+1次方+12ax的n次方
=﹣2ax的n-1次方（9x²-6x+1）