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a)\(M\left(x\right)=3x^4-x^3-2x^2+5x+7\)
\(N\left(x\right)=-3x^4+x^3+10x^2+x-7\)
a, A(x) = -4x5 - x3 + 42 + 5x + 7 + 4x5 - 6x2
= ( 4x5 - 4x5) - x3 + ( 4x2 - 6x2) + 5x + 7
= -x3 - 2x2 +5x +7
B(x) = -3x4 - 4x3 + 10x2 - 8x + 5x3 -7 +8x
= -3x4 + ( 5x3 - 4x3 ) + 10x2 + ( 8x - 8x )
= -3x4 + x3 + 10x2
b, A(x) = -x3 - 2x2 + 5x +7
+
B(x) = -3x4 + x3 + 10x2
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P(x) = A(x) +B(x) = -3x4 + 8x2 + 5x + 7
A(x) = -x3 - 2x2 + 5x + 7
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B(x) = -3x4 + x3 + 10x2
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Q(x) = A(x) - B(x) = 3x4 - 2x3 - 12x2 + 5x + 7
a. Rút gọn đa thức và sắp xếp theo thứ tự giảm dần của biến..
\(A\left(x\right)=13x^4+3x^2+15x+7x^2-10x^4-7x-6-8x+15\)
\(=\left(13x^4-10x^4\right)+\left(3x^2+7x^2\right)+\left(15x-7x-8x\right)+\left(15-6\right)\)
\(=3x^4+10x^2+9.\)
\(B\left(x\right)=5x^4+10-5x^2-18+3x-10x^2-3x-4x^4\)
\(=\left(5x^4-4x^4\right)+\left(-5x^2-3x^2\right)+\left(3x-3x\right)+\left(10-18\right)\)
\(=x^4-8x^2-8\)
b. Tính M = A(x) + B(x) ; N = A(x) - B(x)
\(M=A\left(x\right)+B\left(x\right)=\left(3x^4+10x^2+9\right)+\left(x^4-8x^2-8\right)\)
\(=\left(3x^4+x^4\right)+\left(10x^2-8x^2\right)+\left(10-8\right)\)
\(=4x^4+2x^2+2\)
\(N=A\left(x\right)-B\left(x\right)=\left(3x^4+10x^2+9\right)-\left(x^4-8x^2-8\right)\)
\(=3x^4+10x^2+9-x^4+8x^2+8\)
\(=\left(3x^4-x^4\right)+\left(10x^2+8x^2\right)+\left(9+8\right)\)
\(=2x^4+18x^2+17\)
a. M(x) + N(x) = 3x3 - 3x + x2 + 5 + 2x2 - x + 3x3 + 9
= (3x3 + 3x3) + ( x2 + 2x2 ) + ( -3x - x ) + (5 + 9)
= 6x3 + 3x2 - 4x + 14
b. M(x) + N(x) - P(x) = 6x3 + 3x2 + 2x
=> 6x3 + 3x2 - 4x + 14 - P(x) = 6x3 + 3x2 + 2x
=> 6x3 + 3x2 - 4x + 14 - ( 6x3 + 3x2 + 2x) = P(x)
=> 6x3 + 3x2 - 4x + 14 - 6x3 - 3x2 - 2x = P(x)
=> (6x3 - 6x3 ) + (3x2 - 3x2 ) + (-4x - 2x ) + 14 = P(x)
=> -6x + 14 = P(x)
Ta có : -6x + 14 = 0
=> -6x = -14
=> x = 7/3
=> Đa thức P(x) = -6x + 14 có nghiệm là 7/3
=>
a) Ta có: \(A\left(x\right)-B\left(x\right)\) \(=\left(-5^3+3x^4+\dfrac{2}{7}-8x^2-10x\right)-\left(-2x^4-\dfrac{3}{7}+7x^2+8x^3+6x\right)\)
\(=-5^3+3x^4+\dfrac{2}{7}-8x^2-10x+2x^4+\dfrac{3}{7}-7x^2-8x^3-6x\)
\(=-5^3+\left(3x^4+2x^4\right)+\left(\dfrac{2}{7}+\dfrac{3}{7}\right)-\left(8x^2+7x^2\right)-\left(10x+6x\right)\)
\(=-125+5x^4-15x^2-16x+\dfrac{5}{7}\)
b) Lại có: \(M\left(x\right)-A\left(x\right)=B\left(x\right)\)
\(\Rightarrow M\left(x\right)=B\left(x\right)+A\left(x\right)\)
\(\Rightarrow M\left(x\right)=\left(-5^3+3x^4+\dfrac{2}{7}-8x^2-10x\right)+\left(-2x^4-\dfrac{3}{7}+7x^2+8x^3+6x\right)\)
\(\Rightarrow M\left(x\right)=-5^3+3x^4+\dfrac{2}{7}-8x^2-10x-2x^4-\dfrac{3}{7}+7x^2+8x^3+6x\)
\(\Rightarrow M\left(x\right)=-5^3+\left(3x^4-2x^4\right)+\left(\dfrac{2}{7}-\dfrac{3}{7}\right)-\left(8x^2-7x^2\right)-\left(10x-6x\right)\)
\(\Rightarrow M\left(x\right)=-125+x^4-x^2-4x-\dfrac{1}{7}\)
Vậy .....
\(a) M(x)+N(x)=5x^3-10x^2-8x+10+3x^3+6x^2-3x+7\\ =(5x^3+3x^3)+(-10x^2+6x^2)+(-8x-3x)+(10+7)\\ =8x^3-4x^2-11x+17\\\\ b) M(x)-N(x)=(5x^3-10x^2-8x+10)-(3x^3+6x^2-3x+7)\\ =5x^3-10x^2-8x+10-3x^3-6x^2+3x-7\\ =(5x^3-3x^3)+(-10x^2-6x^2)+(-8x+3x)+(10-7)\\ =2x^3-16x^2-5x+3\)