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Ta có :
A(x) = 3x - 2x2 - 2 +6x2 = 4x2 + 3x - 2
B(x) = 3x2 - x - 2x3 + 4 = -2x3 + 3x2 - x + 4
C(x) = 1 + 4x3 - 2x = 4x3 - 2x + 1
⇒ A(x) + B(x) - C(x)
= (4x2 + 3x - 2) + (-2x3 + 3x2 - x + 4) - (4x3 - 2x + 1)
= 4x2 + 3x - 2 - 2x3 + 3x2 - x + 4 - 4x3 + 2x - 1
= 7x2 + 4x + 1 - 6x3 = -6x3 + 7x2 + 4x + 1
Lời giải:
a)
$M(x)=(x^5+5x^5)-2x^4-4x^3+3x$
$=6x^5-2x^4-4x^3+3x$
$N(x)=-6x^5+(7x^4-5x^4)+(x^3+3x^3)+4x^2-3x-1$
$=-6x^5+2x^4+4x^3+4x^2-3x-1$
b)
$M(-1)=6(-1)^5-2(-1)^4-4(-1)^3+3(-1)=-7$
$N(-2)=-6(-2)^5+2(-2)^4+4(-2)^3+4(-2)^2-3(-2)-1$
$=213$
c)
$M(x)+N(x)=(6x^5-2x^4-4x^3+3x)+(-6x^5+2x^4+4x^3+4x^2-3x-1)$
$=4x^2-1$
$M(x)-N(x)=(6x^5-2x^4-4x^3+3x)-(-6x^5+2x^4+4x^3+4x^2-3x-1)$
$=12x^5-4x^4-8x^3-4x^2+6x+1$
d)
$F(x)=M(x)+N(x)=4x^2-1=0\Leftrightarrow x^2=\frac{1}{4}$
$\Leftrightarrow x=\pm \frac{1}{2}$
Vậy $x=\pm \frac{1}{2}$ là nghiệm của $F(x)$
Dạng 1:
a) $4x+9=4x+\frac{9}{4}.4=4(x+\frac{9}{4}\Rightarrow$ Nghiệm là $-\frac{9}{4}$
b) $-5x+6=-5x+(-5).(-\frac{6}{5})=-5(x-\frac{6}{5})\Rightarrow$ Nghiệm là $\frac{6}{5}$
c) $7-2x=-2x+7=-2x+(-2).(-\frac{7}{2})=-2(x-\frac{7}{2})\Rightarrow$ Nghiệm là $\frac{7}{2}$
d) $2x+5=2x+2.\frac{5}{2}=2.(x+\frac{5}{2})\Rightarrow$ Nghiệm là $-\frac{5}{2}$
e) $2x+6=2x+2.3=2(x+3)\Rightarrow$ Nghiệm là -3
g) $3x-\frac{1}{4}=3x-3.(\frac{1}{12})=3(x-\frac{1}{12})\Rightarrow$ Nghiệm là $\frac{1}{12}$
h) $3x-9=3x-3.3=3(x-3)\Rightarrow$ Nghiệm là 3
k) $-3x-\frac{1}{2}=-3x-3.(\frac{1}{6})=-3(x+\frac{1}{6})\Rightarrow$ Nghiệm là $-\frac{1}{6}$
m) $-17x-34=-17x-17.2=-17(x+2)\Rightarrow$ Nghiệm là -2
n) $2x-1=2x+2.(-\frac{1}{2})=3(x-\frac{1}{2})\Rightarrow$ Nghiệm là $\frac{1}{2}$
q) $5-3x=-3x+5=-3x+(-3).(-\frac{5}{3})=-3(x-\frac{5}{3})\Rightarrow$ Nghiệm là $\frac{5}{3}$
p) $3x-6=3x+3.(-2)=3(x-2)\Rightarrow$ Nghiệm là 2
a)\(A\left(x\right)=x^4+4x^3+2x^2+x-7\)
\(B\left(x\right)=2x^4-4x^3-2x^2-5x+3\)
b) \(f\left(x\right)=A\left(x\right)+B\left(x\right)=x^4+4x^3+2x^2+x-7+2x^4-4x^3-2x^2-5x+3=3x^4-4x-4\)
\(g\left(x\right)=A\left(x\right)-B\left(x\right)=x^4+4x^3+2x^2+x-7-2x^4+4x^3+2x^2+5x-3=-x^4+8x^3+4x^2+6x-10\)c)\(g\left(0\right)=-0^4+8.0^3+4.0^2+6.0-10=-10\)
\(g\left(-2\right)=\left(-2\right)^4+8.\left(-2\right)^3+4.\left(-2\right)^2+6.\left(-2\right)-10=16-64+16-12-10=-54\)
a/ Thu gọn và sắp xếp:
\(P\left(x\right)=x^2+5x^4-3x^3+x^2+4x^4+3x^3-x+5=\left(5x^4+4x^4\right)+\left(3x^3-3x^3\right)+\left(x^2+x^2\right)-x+5=9x^4+2x^2-x+5\)
---
\(Q\left(x\right)=x-5x^3-x^2-x^4+4x^3-x^2+3x-1=-x^4+\left(4x^3-5x^3\right)+\left(-x^2-x^2\right)+\left(x+3x\right)-1=-x^4-x^3-2x^2+4x-1\)
b/ \(P\left(x\right)+Q\left(x\right)=9x^4+2x^2-x+5+\left(-x^4-x^3-2x^2+4x-1\right)=9x^4+2x^2-x+5-x^4-x^3-2x^2+4x-1=8x^4-x^3+3x+4\)
--
\(P\left(x\right)-Q\left(x\right)=9x^4+2x^2-x+5-\left(-x^4-x^3-2x^2+4x-1\right)=9x^4+2x^2-x+5+x^4+x^3+2x^2-4x+1=10x^4+x^3+4x^2-5x+6\)
Câu a họ bảo: thu gọn, sắp xếp theo lũy thừa giảm của P(x) cũng là thu gọn, sắp xếp theo lũy thừa giảm của biến à cậu?
a) Thu gọn, sắp xếp các đa thức theo lũy thừa tăng của biến
f(x)=x2+2x3−7x5−9−6x7+x3+x2+x5−4x2+3x7
= -9 - 2x2 + 3x3 - 6x5 - 3x7
g(x)=x5+2x3−5x8−x7+x3+4x2−5x7+x4−4x2−x6−12
= -12 + 3x3 + x4 + x5 - x6 - 6x7 - 5x8
h(x)=x+4x5−5x6−x7+4x3+x2−2x7+x6−4x2−7x7+x
= 2x - 3x2 + 4x3 +4x5 -4x6 - 10x7
b) Tính f(x) + g(x) − h(x) = ( -9 - 2x2 + 3x3 - 6x5 - 3x7 ) + (-12 + 3x3 + x4 + x5 - x6 - 6x7 - 5x8 ) - (2x - 3x2 + 4x3 +4x5 -4x6 - 10x7)
= - 9 - 2x2 + 3x3 - 6x5 - 3x7 -12 + 3x3 + x4 + x5 - x6 - 6x7 - 5x8 - 2x + 3x2 - 4x3 - 4x5 + 4x6 + 10x7
= -21 - 2x + x2 + 2x3 + x4 - 9x5 + 3x6 + x7 - 5x8
Ta có: A(x) = -4x5 - x3 + 4x2 + 5x + 9 + 4x5 - 6x2 - 2
A(x) = (-4x5 + 4x5) - x3 + (4x2 - 6x2) + 5x + (9 - 2)
A(x) = -x3 - 2x2 + 5x + 7
B(x) = -3x4 - 2x3 + 10x2 - 8x + 5x3 - 7 - 2x3 + 8x
B(x) = -3x4 - (2x3 - 5x3 + 2x3) + 10x2 - (8x - 8x) - 7
B(x) = -3x4 + x3 + 10x2 - 7
A(x) + B(x) = (-x3 - 2x2 + 5x + 7) + (-3x4 + x3 + 10x2 - 7)
= -x3 - 2x2 + 5x + 7 - 3x4 + x3 + 10x2 - 7
= (-x3 + x3) - (2x2 - 10x2) + 5x + (7 - 7)
= 8x2 + 5x
A(x) - B(x) = (-x^3 - 2x^2 + 5x + 7) - (-3x^4 + x^3 + 10x^2 - 7)
= -x^3 - 2x^2 + 5x + 7 + 3x^4 - x^3 - 10x^2 + 7
= (-x^3 - x^3) - (2x^2 + 10x^2) + 5x + (7 + 7)
= -2x^3 - 12x^2 + 5x + 14
f(x)=\(9-x^5-7x^4-2x^3+x^2+4x\)
g(x)=\(x^5-7x^4+4x^3-3x-9\)
f(x)+g(x)=\(9-x^5-7x^4-2x^3+x^2+4x\)+\(x^5-7x^4+4x^3-3x-9\)
=(9-9)-(\(x^5-x^5\))\(-\left(7x^4+7x^4\right)-\left(2x^3-4x^3\right)+x^2\)+(\(\)\(4x-3x\))
=\(-14x^4+2x^3+x^2+x\)
a) Sắp xếp các đa thức theo lũy thừa giảm của biến :
\(f\left(x\right)=-x^5-7x^4-2x^3+x^2+4x+9\)
\(g\left(x\right)=x^5-7x^4+2x^3+2x^3-3x-9\)
b, \(h\left(x\right)=f\left(x\right)+g\left(x\right)\)
\(=\left(-x^5-7x^4-2x^3+x^2+4x+9\right)+\left(x^5-7x^4+2x^3+2x^3-3x-9\right)\)
=> h(x) = -14x4 + 2x3 + x2 +x
\(A\left(x\right)=5x^3+3x^2-x-7\)
\(B\left(x\right)=7x^3-3x+4\)
=>\(5x^3+3x^2-x-7=7x^3-3x+4\)
\(\Leftrightarrow-2x^3+3x^2+2x-11=0\)
hay \(x\in\left\{-1.52\right\}\)
1: \(A\left(x\right)=6x^4-5x^2+4x-3x^4+2x^3\)
\(=\left(6x^4-3x^4\right)+\left(2x^3-5x^2+4x\right)\)
\(=3x^4+2x^3-5x^2+4x\)
2: \(A\left(x\right)=3x^2+7x^3-3x^3+6x^3-3x^2\)
\(=\left(7x^3-3x^3+6x^3\right)+\left(3x^2-3x^2\right)\)
\(=10x^3\)
3: \(A\left(x\right)=x^5+x^4-3x+7-2x^4-x^5\)
\(=\left(x^5-x^5\right)+\left(x^4-2x^4\right)-3x+7\)
\(=-x^4-3x+7\)
4: \(A\left(x\right)=1-6x^7+5x^4-2+13x^5-8x^7\)
\(=\left(-6x^7-8x^7\right)+13x^5+5x^4+\left(1-2\right)\)
\(=-14x^7+13x^5+5x^4-1\)
5: \(A\left(x\right)=3x^2-2x+7+2x-3x^2-6\)
\(=\left(3x^2-3x^2\right)+\left(-2x+2x\right)+\left(7-6\right)\)
=1
6: \(A\left(x\right)=2-9x^2+4x^5-3x^3+x-4x^5\)
\(=\left(4x^5-4x^5\right)+\left(-3x^3\right)+\left(-9x^2+x+2\right)\)
\(=-3x^3-9x^2+x+2\)
7: \(A\left(x\right)=2x^3+5-7x^4-6x^3+3x^2-x^5\)
\(=-x^5-7x^4+\left(2x^3-6x^3\right)+3x^2+5\)
\(=-x^5-7x^4-4x^3+3x^2+5\)
8: \(A\left(x\right)=4x^5+3x-2x^2-x^5+4x^2-8\)
\(=\left(4x^5-x^5\right)+\left(4x^2-2x^2\right)+3x-8\)
\(=3x^5+2x^2+3x-8\)
9: \(A\left(x\right)=2+5x^2-3x^3+4x^2-2x-x^3\)
\(=\left(-3x^3-x^3\right)+\left(5x^2+4x^2\right)+\left(-2x+2\right)\)
\(=-4x^3+9x^2-2x+2\)
10: \(A\left(x\right)=-6x^4+2x^3+x+5x^4-2x+3x^3\)
\(=\left(-6x^4+5x^4\right)+\left(2x^3+3x^2\right)+\left(x-2x\right)\)
\(=-x^4+5x^2-x\)
1) \(A \left(\right. x \left.\right) = 6 x^{4} - 5 x^{2} + 4 x - 3 x^{4} + 2 x^{3}\)
\(A \left(\right. x \left.\right) = \left(\right. 6 x^{4} - 3 x^{4} \left.\right) + 2 x^{3} - 5 x^{2} + 4 x\) \(A \left(\right. x \left.\right) = 3 x^{4} + 2 x^{3} - 5 x^{2} + 4 x\)
2) \(A \left(\right. x \left.\right) = 3 x^{2} + 7 x^{3} - 3 x^{3} + 6 x^{3} - 3 x^{2}\)
\(A \left(\right. x \left.\right) = \left(\right. 7 x^{3} - 3 x^{3} + 6 x^{3} \left.\right) + \left(\right. 3 x^{2} - 3 x^{2} \left.\right)\) \(A \left(\right. x \left.\right) = 10 x^{3} + 0 x^{2}\)
3) \(A \left(\right. x \left.\right) = x^{5} + x^{4} - 3 x + 7 - 2 x^{4} - x^{5}\)
\(A \left(\right. x \left.\right) = \left(\right. x^{5} - x^{5} \left.\right) + \left(\right. x^{4} - 2 x^{4} \left.\right) - 3 x + 7\) \(A \left(\right. x \left.\right) = 0 x^{5} - x^{4} - 3 x + 7\)
4) \(A \left(\right. x \left.\right) = 1 - 6 x^{7} + 5 x^{4} - 2 + 13 x^{5} - 8 x^{7}\)
\(A \left(\right. x \left.\right) = \left(\right. 1 - 2 \left.\right) + \left(\right. - 6 x^{7} - 8 x^{7} \left.\right) + 13 x^{5} + 5 x^{4}\) \(A \left(\right. x \left.\right) = - 1 - 14 x^{7} + 13 x^{5} + 5 x^{4}\)
5) \(A \left(\right. x \left.\right) = 3 x^{2} - 2 x + 7 + 2 x - 3 x^{2} - 6\)
\(A \left(\right. x \left.\right) = \left(\right. 3 x^{2} - 3 x^{2} \left.\right) + \left(\right. - 2 x + 2 x \left.\right) + \left(\right. 7 - 6 \left.\right)\) \(A \left(\right. x \left.\right) = 0 x^{2} + 0 x + 1\)
6) \(A \left(\right. x \left.\right) = 2 - 9 x^{2} + 4 x^{5} - 3 x^{3} + x - 4 x^{5}\)
\(A \left(\right. x \left.\right) = \left(\right. 4 x^{5} - 4 x^{5} \left.\right) - 9 x^{2} - 3 x^{3} + x + 2\) \(A \left(\right. x \left.\right) = 0 x^{5} - 9 x^{2} - 3 x^{3} + x + 2\)
7) \(A \left(\right. x \left.\right) = 2 x^{3} + 5 - 7 x^{4} - 6 x^{3} + 3 x^{2} - x^{5}\)
\(A \left(\right. x \left.\right) = \left(\right. - x^{5} - 7 x^{4} + 2 x^{3} - 6 x^{3} + 3 x^{2} + 5 \left.\right)\) \(A \left(\right. x \left.\right) = - x^{5} - 7 x^{4} - 4 x^{3} + 3 x^{2} + 5\)
8) \(A \left(\right. x \left.\right) = 4 x^{5} + 3 x - 2 x^{2} - x^{5} + 4 x^{2} - 8\)
\(A \left(\right. x \left.\right) = \left(\right. 4 x^{5} - x^{5} \left.\right) + \left(\right. 4 x^{2} - 2 x^{2} \left.\right) + 3 x - 8\) \(A \left(\right. x \left.\right) = 3 x^{5} + 2 x^{2} + 3 x - 8\)
9) \(A \left(\right. x \left.\right) = 2 + 5 x^{2} - 3 x^{3} + 4 x^{2} - 2 x - x^{3}\)
\(A \left(\right. x \left.\right) = 2 + \left(\right. 5 x^{2} + 4 x^{2} \left.\right) + \left(\right. - 3 x^{3} - x^{3} \left.\right) - 2 x\) \(A \left(\right. x \left.\right) = 2 + 9 x^{2} - 4 x^{3} - 2 x\)
10) \(A \left(\right. x \left.\right) = - 6 x^{4} + 2 x^{3} + x + 5 x^{4} - 2 x + 3 x^{3}\)
\(A \left(\right. x \left.\right) = \left(\right. - 6 x^{4} + 5 x^{4} \left.\right) + \left(\right. 2 x^{3} + 3 x^{3} \left.\right) + \left(\right. x - 2 x \left.\right)\) \(A \left(\right. x \left.\right) = - x^{4} + 5 x^{3} - x\)
Hy vọng bạn đã hiểu cách thu gọn và sắp xếp các biểu thức đại số này!