Упр.360 ГДЗ Никольский Потапов 7 класс (Алгебра)
Преобразуйте выражение в многочлен стандартного вида:
а) $$\left(m+n\right)^2+\left(m-n\right)^2$$;
б) $$2\left(a-1\right)^2+3\left(a-2\right)^2$$;
в) $$5\left(x-y\right)^2+\left(x-2y\right)^2$$;
г) $$4\left(m-2n\right)^2-3\left(3m+n\right)^2$$;
д) $$3\left(2a-b\right)^2-5\left(a-2b\right)^2$$;
е) $$4\left(3x+4y\right)^2-7\left(2x-3y\right)^2$$;
ж) $$2\left(p-3q\right)^2-4\left(2p-q\right)^2-\left(2q-3p\right)\left(p+q\right)$$;
з) $$5\left(n-5m\right)^2-6\left(2n-3m\right)^2-\left(3m-n\right)\left(7m-n\right)$$;
и) $$\left(2p-q\right)^2-2\left(2p-q\right)\left(p-q\right)+\left(p-q\right)^2$$.
а) Раскроем скобки и приведём подобные:
$$(m+n)^2+(m-n)^2=m^2+2mn+n^2+m^2-2mn+n^2=2m^2+2n^2.$$
б)
$$2(a-1)^2+3(a-2)^2=2(a^2-2a+1)+3(a^2-4a+4)$$
$$=2a^2-4a+2+3a^2-12a+12=5a^2-16a+14.$$
в)
$$5(x-y)^2+(x-2y)^2=5(x^2-2xy+y^2)+x^2-4xy+4y^2$$
$$=5x^2-10xy+5y^2+x^2-4xy+4y^2=6x^2-14xy+9y^2.$$
г)
$$4(m-2n)^2-3(3m+n)^2=4(m^2-4mn+4n^2)-3(9m^2+6mn+n^2)$$
$$=4m^2-16mn+16n^2-27m^2-18mn-3n^2 =13n^2-34mn-23m^2.$$
д)
$$3(2a-b)^2-5(a-2b)^2=3(4a^2-4ab+b^2)-5(a^2-4ab+4b^2)$$
$$=12a^2-12ab+3b^2-5a^2+20ab-20b^2=7a^2+8ab-17b^2.$$
е)
$$4(3x+4y)^2-7(2x-3y)^2=4(9x^2+24xy+16y^2)-7(4x^2-12xy+9y^2)$$
$$=36x^2+96xy+64y^2-28x^2+84xy-63y^2=8x^2+180xy+y^2.$$
ж)
$$2(p-3q)^2-4(2p-q)^2-(2q-3p)(p+q)$$
$$=2(p^2-6pq+9q^2)-4(4p^2-4pq+q^2)-(2pq+2q^2-3p^2-3pq)$$
$$=2p^2-12pq+18q^2-16p^2+16pq-4q^2+3p^2+pq-2q^2 =12q^2+5pq-11p^2.$$
з)
$$5(n-5m)^2-6(2n-3m)^2-(3m-n)(7m-n)$$
$$=5(n^2-10mn+25m^2)-6(4n^2-12mn+9m^2)-(21m^2-10mn+n^2)$$
$$=5n^2-50mn+125m^2-24n^2+72mn-54m^2-21m^2+10mn-n^2 =50m^2+32mn-20n^2.$$
и)
$$(2p-q)^2-2(2p-q)(p-q)+(p-q)^2$$
$$=4p^2-4pq+q^2-2(2p^2-3pq+q^2)+p^2-2pq+q^2$$
$$=4p^2-4pq+q^2-4p^2+6pq-2q^2+p^2-2pq+q^2=p^2.$$
Ответ
а) $$2m^2+2n^2$$; б) $$5a^2-16a+14$$; в) $$6x^2-14xy+9y^2$$; г) $$13n^2-34mn-23m^2$$; д) $$7a^2+8ab-17b^2$$; е) $$8x^2+180xy+y^2$$; ж) $$12q^2+5pq-11p^2$$; з) $$50m^2+32mn-20n^2$$; и) $$p^2$$.
