解決(x-(-6-i))(x-(-6+i))(x-(-1+3i))(x-(-1+3i)) |Microsoft數學求解器

$$( x - ( - 6 - i ) ) ( x - ( - 6 + i ) ) ( x - ( - 1 + 3 i ) ) ( x - ( - 1 + 3 i ) )$$

評估

$$\left(x+\left(6-i\right)\right)\left(x+\left(6+i\right)\right)\left(x+\left(1-3i\right)\right)^{2}$$

解題步驟

將 x-\left(-1+3i\right) 乘上 x-\left(-1+3i\right) 得到 \left(x-\left(-1+3i\right)\right)^{2}。

$$\left(x-\left(-6-i\right)\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

-6-i 的相反數是 6+i。

$$\left(x+\left(6+i\right)\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

計算 x+\left(6+i\right) 乘上 x-\left(-6+i\right) 時使用乘法分配律。

$$\left(x\left(x-\left(-6+i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

計算 x\left(x-\left(-6+i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right) 乘上 \left(x-\left(-1+3i\right)\right)^{2} 時使用乘法分配律。

$$x\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -6+i 得到 6-i。

$$x\left(x+\left(6-i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -1+3i 得到 1-3i。

$$x\left(x+\left(6-i\right)\right)\left(x+\left(1-3i\right)\right)^{2}+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

使用二項式定理 \left(a+b\right)^{2}=a^{2}+2ab+b^{2} 展開 \left(x+\left(1-3i\right)\right)^{2}。

$$x\left(x+\left(6-i\right)\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

計算 x 乘上 x+\left(6-i\right) 時使用乘法分配律。

$$\left(x^{2}+\left(6-i\right)x\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

透過將 x^{2}+\left(6-i\right)x 的每個項乘以 x^{2}+\left(2-6i\right)x+\left(-8-6i\right) 的每個項以套用乘法分配律。

$$x^{4}+\left(2-6i\right)x^{3}+\left(-8-6i\right)x^{2}+\left(6-i\right)x^{3}+\left(6-38i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

合併 \left(2-6i\right)x^{3} 和 \left(6-i\right)x^{3} 以取得 \left(8-7i\right)x^{3}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-8-6i\right)x^{2}+\left(6-38i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

合併 \left(-8-6i\right)x^{2} 和 \left(6-38i\right)x^{2} 以取得 \left(-2-44i\right)x^{2}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -6+i 得到 6-i。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x+\left(6-i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -1+3i 得到 1-3i。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x+\left(6-i\right)\right)\left(x+\left(1-3i\right)\right)^{2}$$

使用二項式定理 \left(a+b\right)^{2}=a^{2}+2ab+b^{2} 展開 \left(x+\left(1-3i\right)\right)^{2}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x+\left(6-i\right)\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)$$

計算 6+i 乘上 x+\left(6-i\right) 時使用乘法分配律。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(\left(6+i\right)x+37\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)$$

透過將 \left(6+i\right)x+37 的每個項乘以 x^{2}+\left(2-6i\right)x+\left(-8-6i\right) 的每個項以套用乘法分配律。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)x^{3}+\left(18-34i\right)x^{2}+\left(-42-44i\right)x+37x^{2}+\left(74-222i\right)x+\left(-296-222i\right)$$

合併 \left(18-34i\right)x^{2} 和 37x^{2} 以取得 \left(55-34i\right)x^{2}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)x^{3}+\left(55-34i\right)x^{2}+\left(-42-44i\right)x+\left(74-222i\right)x+\left(-296-222i\right)$$

合併 \left(-42-44i\right)x 和 \left(74-222i\right)x 以取得 \left(32-266i\right)x。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)x^{3}+\left(55-34i\right)x^{2}+\left(32-266i\right)x+\left(-296-222i\right)$$

合併 \left(8-7i\right)x^{3} 和 \left(6+i\right)x^{3} 以取得 \left(14-6i\right)x^{3}。

$$x^{4}+\left(14-6i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(55-34i\right)x^{2}+\left(32-266i\right)x+\left(-296-222i\right)$$

合併 \left(-2-44i\right)x^{2} 和 \left(55-34i\right)x^{2} 以取得 \left(53-78i\right)x^{2}。

$$x^{4}+\left(14-6i\right)x^{3}+\left(53-78i\right)x^{2}+\left(-54-28i\right)x+\left(32-266i\right)x+\left(-296-222i\right)$$

合併 \left(-54-28i\right)x 和 \left(32-266i\right)x 以取得 \left(-22-294i\right)x。

$$x^{4}+\left(14-6i\right)x^{3}+\left(53-78i\right)x^{2}+\left(-22-294i\right)x+\left(-296-222i\right)$$

展開

$$x^{4}+\left(14-6i\right)x^{3}+\left(53-78i\right)x^{2}+\left(-22-294i\right)x+\left(-296-222i\right)$$

解題步驟

將 x-\left(-1+3i\right) 乘上 x-\left(-1+3i\right) 得到 \left(x-\left(-1+3i\right)\right)^{2}。

$$\left(x-\left(-6-i\right)\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

-6-i 的相反數是 6+i。

$$\left(x+\left(6+i\right)\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

計算 x+\left(6+i\right) 乘上 x-\left(-6+i\right) 時使用乘法分配律。

$$\left(x\left(x-\left(-6+i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

計算 x\left(x-\left(-6+i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right) 乘上 \left(x-\left(-1+3i\right)\right)^{2} 時使用乘法分配律。

$$x\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -6+i 得到 6-i。

$$x\left(x+\left(6-i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -1+3i 得到 1-3i。

$$x\left(x+\left(6-i\right)\right)\left(x+\left(1-3i\right)\right)^{2}+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

使用二項式定理 \left(a+b\right)^{2}=a^{2}+2ab+b^{2} 展開 \left(x+\left(1-3i\right)\right)^{2}。

$$x\left(x+\left(6-i\right)\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

計算 x 乘上 x+\left(6-i\right) 時使用乘法分配律。

$$\left(x^{2}+\left(6-i\right)x\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

透過將 x^{2}+\left(6-i\right)x 的每個項乘以 x^{2}+\left(2-6i\right)x+\left(-8-6i\right) 的每個項以套用乘法分配律。

$$x^{4}+\left(2-6i\right)x^{3}+\left(-8-6i\right)x^{2}+\left(6-i\right)x^{3}+\left(6-38i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

合併 \left(2-6i\right)x^{3} 和 \left(6-i\right)x^{3} 以取得 \left(8-7i\right)x^{3}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-8-6i\right)x^{2}+\left(6-38i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

合併 \left(-8-6i\right)x^{2} 和 \left(6-38i\right)x^{2} 以取得 \left(-2-44i\right)x^{2}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x-\left(-6+i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -6+i 得到 6-i。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x+\left(6-i\right)\right)\left(x-\left(-1+3i\right)\right)^{2}$$

將 -1 乘上 -1+3i 得到 1-3i。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x+\left(6-i\right)\right)\left(x+\left(1-3i\right)\right)^{2}$$

使用二項式定理 \left(a+b\right)^{2}=a^{2}+2ab+b^{2} 展開 \left(x+\left(1-3i\right)\right)^{2}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)\left(x+\left(6-i\right)\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)$$

計算 6+i 乘上 x+\left(6-i\right) 時使用乘法分配律。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(\left(6+i\right)x+37\right)\left(x^{2}+\left(2-6i\right)x+\left(-8-6i\right)\right)$$

透過將 \left(6+i\right)x+37 的每個項乘以 x^{2}+\left(2-6i\right)x+\left(-8-6i\right) 的每個項以套用乘法分配律。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)x^{3}+\left(18-34i\right)x^{2}+\left(-42-44i\right)x+37x^{2}+\left(74-222i\right)x+\left(-296-222i\right)$$

合併 \left(18-34i\right)x^{2} 和 37x^{2} 以取得 \left(55-34i\right)x^{2}。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)x^{3}+\left(55-34i\right)x^{2}+\left(-42-44i\right)x+\left(74-222i\right)x+\left(-296-222i\right)$$

合併 \left(-42-44i\right)x 和 \left(74-222i\right)x 以取得 \left(32-266i\right)x。

$$x^{4}+\left(8-7i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(6+i\right)x^{3}+\left(55-34i\right)x^{2}+\left(32-266i\right)x+\left(-296-222i\right)$$

合併 \left(8-7i\right)x^{3} 和 \left(6+i\right)x^{3} 以取得 \left(14-6i\right)x^{3}。

$$x^{4}+\left(14-6i\right)x^{3}+\left(-2-44i\right)x^{2}+\left(-54-28i\right)x+\left(55-34i\right)x^{2}+\left(32-266i\right)x+\left(-296-222i\right)$$

合併 \left(-2-44i\right)x^{2} 和 \left(55-34i\right)x^{2} 以取得 \left(53-78i\right)x^{2}。

$$x^{4}+\left(14-6i\right)x^{3}+\left(53-78i\right)x^{2}+\left(-54-28i\right)x+\left(32-266i\right)x+\left(-296-222i\right)$$

合併 \left(-54-28i\right)x 和 \left(32-266i\right)x 以取得 \left(-22-294i\right)x。

$$x^{4}+\left(14-6i\right)x^{3}+\left(53-78i\right)x^{2}+\left(-22-294i\right)x+\left(-296-222i\right)$$