397° Вправа 397Алгебра 8 клас Тарасенкова 2025 · § 10 · стор. 110

397. Розкладіть на множники:

${\displaystyle A^2-B^2=}$${\displaystyle (A-B)(A+B)}$.

1) ${\displaystyle x^{-2}-25=}$${\displaystyle {\left(x^{-1}\right)}^2-5^2=}$${\displaystyle \left(x^{-1}-5\right)\left(x^{-1}+5\right)}$;

2) ${\displaystyle a^{-2}-b^2=}$${\displaystyle {\left(a^{-1}\right)}^2-b^2=}$${\displaystyle \left(a^{-1}-b\right)\left(a^{-1}+b\right)}$;

3) ${\displaystyle n^{-2}-m^{-2}=}$${\displaystyle {\left(n^{-1}\right)}^2-{\left(m^{-1}\right)}^2=}$${\displaystyle \left(n^{-1}-m^{-1}\right)\left(n^{-1}+m^{-1}\right)}$;

4) ${\displaystyle x^{-8}-{(1+x^{-4})}^2=}$${\displaystyle {\left(x^{-4}\right)}^2-{(1+x^{-4})}^2=}$${\displaystyle (\cancel{x^{-4}}-1\cancel{-x^{-4}})(x^{-4}+1+x^{-4})=}$${\displaystyle -\left(2x^{-4}+1\right)}$;

5) ${\displaystyle {(n^{-2}-m^{-2})}^2-{(m^{-2}+n^{-2})}^2=}$${\displaystyle (\cancel{n^{-2}}-m^{-2}-m^{-2}\cancel{-n^{-2}})(n^{-2}\cancel{-m^{-2}}\cancel{+m^{-2}}+n^{-2})=}$${\displaystyle (-2m^{-2})\cdot 2n^{-2}=}$${\displaystyle -4m^{-2}{n}^{-2}}$;

6) ${\displaystyle {(x^{-2}+2)}^2-{(3x^{-2}-4)}^2=}$${\displaystyle (x^{-2}+2-3x^{-2}+4)(x^{-2}+2+3x^{-2}-4)=}$${\displaystyle (6-2x^{-2})(4x^{-2}-2)=}$${\displaystyle 4\left(3-x^{-2}\right)\left(2x^{-2}-1\right)}$.