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20.把下列各式因式分解:
(1)$16x^{4} - 8x^{2}y^{2} + y^{4}$;
(2)$a^{2}(x - y) - 9b^{2}(x - y)$;
(3)$9(3m + 2n)^{2} - 4(m - 2n)^{2}$;
(4)$(a - b)(3a + b)^{2} + (a + 3b)^{2}(b - a)$.
(1)$16x^{4} - 8x^{2}y^{2} + y^{4}$;
(2)$a^{2}(x - y) - 9b^{2}(x - y)$;
(3)$9(3m + 2n)^{2} - 4(m - 2n)^{2}$;
(4)$(a - b)(3a + b)^{2} + (a + 3b)^{2}(b - a)$.
答案：
(1)$16x^{4}-8x^{2}y^{2}+y^{4}=(4x^{2})^{2}-2×4x^{2}\cdot y^{2}+(y^{2})^{2}=(4x^{2}-y^{2})^{2}=(2x+y)^{2}(2x-y)^{2}$.
(2)$a^{2}(x-y)-9b^{2}(x-y)=(x-y)(a^{2}-9b^{2})=(x-y)(a+3b)(a-3b)$.
(3)$9(3m+2n)^{2}-4(m-2n)^{2}=[3(3m+2n)+2(m-2n)]×[3(3m+2n)-2(m-2n)]=(11m+2n)\cdot(7m+10n)$.
(4)$(a-b)(3a+b)^{2}+(a+3b)^{2}(b-a)=(a-b)\cdot[(3a+b)^{2}-(a+3b)^{2}]=(a-b)(3a+b+a+3b)(3a+b-a-3b)=(a-b)(4a+4b)(2a-2b)=8(a-b)(a+b)(a-b)=8(a-b)^{2}(a+b)$.
(1)$16x^{4}-8x^{2}y^{2}+y^{4}=(4x^{2})^{2}-2×4x^{2}\cdot y^{2}+(y^{2})^{2}=(4x^{2}-y^{2})^{2}=(2x+y)^{2}(2x-y)^{2}$.
(2)$a^{2}(x-y)-9b^{2}(x-y)=(x-y)(a^{2}-9b^{2})=(x-y)(a+3b)(a-3b)$.
(3)$9(3m+2n)^{2}-4(m-2n)^{2}=[3(3m+2n)+2(m-2n)]×[3(3m+2n)-2(m-2n)]=(11m+2n)\cdot(7m+10n)$.
(4)$(a-b)(3a+b)^{2}+(a+3b)^{2}(b-a)=(a-b)\cdot[(3a+b)^{2}-(a+3b)^{2}]=(a-b)(3a+b+a+3b)(3a+b-a-3b)=(a-b)(4a+4b)(2a-2b)=8(a-b)(a+b)(a-b)=8(a-b)^{2}(a+b)$.
21.(2025·甘肃陇南期末)已知整式$A = x(x + 3) + 5$,整式$B = ax - 1$.
(1)若$A + B$是完全平方式,求$a$的值;
(2)若$A - B可以分解为(x - 1)(x - 6)$,求$a$.
(1)若$A + B$是完全平方式,求$a$的值;
(2)若$A - B可以分解为(x - 1)(x - 6)$,求$a$.
答案：
(1)$A+B=x(x+3)+5+ax-1=x^{2}+(3+a)x+4$.
∵A+B为完全平方式，
∴$3+a=\pm4$，
∴$a=1$或$-7$.
(2)$A-B=x^{2}+(3-a)x+6$.
∵$(x-1)(x-6)=x^{2}-7x+6$，
∴$3-a=-7$，
∴$a=10$.
(1)$A+B=x(x+3)+5+ax-1=x^{2}+(3+a)x+4$.
∵A+B为完全平方式，
∴$3+a=\pm4$，
∴$a=1$或$-7$.
(2)$A-B=x^{2}+(3-a)x+6$.
∵$(x-1)(x-6)=x^{2}-7x+6$，
∴$3-a=-7$，
∴$a=10$.
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