设x2+ax+b是xn-x3+5x2+x+1与3xn-3x3+14x2+13x+2的公因式,则1+a+b=()
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![设x2+ax+b是xn-x3+5x2+x+1与3xn-3x3+14x2+13x+2的公因式,则1+a+b=()](/uploads/image/z/1269375-15-5.jpg?t=%E8%AE%BEx2%2Bax%2Bb%E6%98%AFxn-x3%2B5x2%2Bx%2B1%E4%B8%8E3xn-3x3%2B14x2%2B13x%2B2%E7%9A%84%E5%85%AC%E5%9B%A0%E5%BC%8F%2C%E5%88%991%2Ba%2Bb%EF%BC%9D%EF%BC%88%EF%BC%89)
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设x2+ax+b是xn-x3+5x2+x+1与3xn-3x3+14x2+13x+2的公因式,则1+a+b=()
设x2+ax+b是xn-x3+5x2+x+1与3xn-3x3+14x2+13x+2的公因式,则1+a+b=()
设x2+ax+b是xn-x3+5x2+x+1与3xn-3x3+14x2+13x+2的公因式,则1+a+b=()
xn-x3+5x2+x+1 (1)
3xn-3x3+14x2+13x+2 (2)
因为x2+ax+b是(1)与(2)的公因式
所以,x2+ax+b是(1)*3 -(2)的因式
(1)*3 -(2)
=3*(xn-x3+5x2+x+1 )-(3xn-3x3+14x2+13x+2)
=15x^2+3x+3-14x^2-13x-2
=x^2-10x+1
x2+ax+b是x^2-10x+1的因式
所以a=-10,b=1
1+a+b=1-10+1=-8
x^2+ax+b是x^n-x^3+5x^2+x+1与3x^n-3x^3+14x^2+13x+2的公因式
(3x^n-3x^3+14x^2+13x+2)-3(x^n-x^3+5x^2+x+1)仍是x^2+ax+b的公因式。
即得-x^2+10x-1=-(x^2-10x+1)
所以1+a+b=1+(-10)+1=-8