多項式の乗法[展開の基本]〜高校数学問題演習編-その2〜

問題

次の式を展開せよ.

解説

• (1)　$2x(2x^{2}-3xy-y^{2})$
  • $2x(2x^{2}-3xy-y^{2})$
    $=$$4x^{3}-6x^{2}y-2xy^{2}$
• (2)　$\bigg(\dfrac{a^{2}}{3}-\dfrac{ab}{6}-\dfrac{b^{2}}{4}\bigg)\times12b^{2}$
  • $\bigg(\dfrac{a^{2}}{3}-\dfrac{ab}{6}-\dfrac{b^{2}}{4}\bigg)\times12b^{2}$
    $=$$4a^{2}b^{2}-2ab^{3}-3b^{4}$
• (3)　$(3x^{2}-4)(2x+5)$
  • $(3x^{2}-4)(2x+5)=3x^{2}\cdot2x+3x^{2}\cdot5-4\cdot2x-4\cdot5$
    $=$$6x^{3}+15x^{2}-8x-20$
• (4)　$(x-1)(x^{2}+2x-3)$
  • $(x-1)(x^{2}+2x-3)$
    $=x\cdot x^{2}+x\cdot2x-x\cdot3-1\cdot x^{2}-1\cdot2x-1\cdot(-3)$
    $=$$x^{3}+x^{2}-5x+3$
• (5)　$(x^{2}-2xy-y^{2})(x-3y)$
  • $(x^{2}-2xy-y^{2})(x-3y)$
    $=x^{2}\cdot x-x^{2}\cdot3y-2xy\cdot x-2xy\cdot(-3y)-y^{2}\cdot x-y^{2}\cdot(-3y)$
    $=$$x^{3}-5x^{2}y+5xy^{2}+3y^{3}$
• (6)　$(3a-4b+2c)(a+2b-5c)$
  • $(3a-4b+2c)(a+2b-5c)$
    $=3a\cdot a+3a\cdot2b-3a\cdot5c-4b\cdot a-4b\cdot2b-4b\cdot(-5c)+2c\cdot a+2c\cdot2b-2c\cdot5c$
    $=$$3a^{2}-8b^{2}-10c^{2}+2ab+24bc-13ca$