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a) 4x2 - 20x + 25 - 36y2
= (2x - 5)2 - 36y2
= (2x - 5 - 6y)(2x - 5 + 6y)
b) x3 + x2 - 2x - 8
= (x3 - 8) + (x2 - 2x)
= (x - 2)(x2 + 2x + 4) + x(x - 2)
= (x - 2)(x2 + 2x + 4 + x)
= (x - 2)(x2 + 3x + 4)
d) x4 + 6x3 + 9x2 - 16
= x2(x2 + 6x + 9) - 16
= x2(x + 3)2 - 16
= (x2 + 3x)2 - 16
= (x2 + 3x - 4)(x2 + 3x + 4)
= (x2 + 4x - x - 4)(x2 + 3x + 4)
= [x(x + 4) - (x + 4)](x2 + 3x + 4)
= (x - 1)(x + 4)(x2 + 3x + 4)
b) x8 +7x4+16
= x8+8x4-x4+16
= (x8+8x4+16) - x4
=(x4+4)2-x4
= (x4+4+x2)(x4+4-x2)
c) x5+x-1
= x5 - x4+x3+x4-x3+x2-x2+x-1
= x3(x2-x+1) + x2(x2-x+1) - (x2-x+1)
= (x2-x+1)(x3+x2 -1)
d)x7+x2+1
=x7-x+x2 +x+1
= x (x6-1) + (x2+x+1)
= x(x3-1)(x3+1) + (x2+x+1)
= x(x3+1)(x-1)(x2+x+1)+(x2+x+1)
= (x2+x+1)[x(x3+1)(x-1) +1]
= (x2+x+1)(x5-x4+x2-x+1)
= x (x-1)(x2+x+1)
e) x5+x4+1
= x5+x4+x3 - x3+1
= x3(x2+x+1) - (x-1)(x2+x+1)
= (x2+x+1)(x3-x+1)
f) x8+x+1
= x8-x2+x2+x+1
= x2(x6-1)+(x2+x+1)
= x2(x3-1)(x3+1) +(x2+x+1)
= (x5+x2)(x-1)(x2+x+1) +(x2+x+1)
= (x2+x+1)(x6-x5+x3-x2+1)
a) x2 + 6x + 9 = x2 + 2 . x . 3 + 32 = (x + 3)2
b) 10x – 25 – x2 = -(-10x + 25 +x2) = -(25 – 10x + x2)
= -(52 – 2 . 5 . x – x2) = -(5 – x)2
c) 8x3 - 1/8 = (2x)3 – (1/2)3 = (2x - 1/2)[(2x)2 + 2x . 12 + (1/2)2]
= (2x - 1/2)(4x2 + x + 1/4)
d)1/25x2 – 64y2 = (1/5x)2(1/5x)2- (8y)2 = (1/5x + 8y)(1/5x - 8y)
1)x2-8x-9
= x^2 - 9x +x -9
= x(x+1) - 9 (x+1)
= (x-9) (x+1)
2)x2+3x-18
3)x3-5x2+4x
=x^3 - 4x^2 - x^2 + 4x
= x^2 (x-1) - 4x(x-1)
= (x^2 - 4x) (x-1)
= x(x-4)(x-1)
4)x3-11x2+30x
5)x3-7x-6
6)x16-64
\(=\left(x^8\right)^2-8^2\)
\(=\left(x^8-8\right)\left(x^8+8\right)\)
7)x3-5x2+8x-4
8)x2-3x+2
= x^2 - 2x - x +2
= x(x-1) -2(x-1)
= (x-2)(x-1)
1) \(\left(x-9\right)\left(x+1\right)\) 2) \(\left(x-3\right)\left(x+6\right)\) 3) \(x\left(x-4\right)\left(x-1\right)\)
4) \(x\left(x-6\right)\left(x-5\right)\) 5)\(\left(x-3\right)\left(x+1\right)\left(x+2\right)\) 6) ........
7) \(\left(x-1\right)\left(x-2\right)\left(x-2\right)\) 8) \(\left(x-2\right)\left(x-1\right)\)
\(a,25-x^4=5^2-\left(x^2\right)^2=\left(5-x^2\right)\left(5+x^2\right)\)
\(b,\left(3x+y\right)^2=\left(3x\right)^2+2.3x.y+y^2=9x^2+6xy+y^2\)
\(c,\left(x+1\right)^2=x^2+2x+1\)
\(d,\left(x+3\right)\left(x^2-3x+9\right)=x^3+27\)
\(e,\left(2+3x\right)^2=2^2+2.2.3x+\left(3x\right)^2=4+12x+9x^2\)
\(f,x^2-9=\left(x-3\right)\left(x+3\right)\)
\(g,\left(x-1\right)^3=x^3-3.x^2.1+3.x.1^2-1^3=x^3-3x^2+3x-1\)
\(h,x^3-8=\left(x-2\right)\left(x^2-2x+4\right)\)
a) \(25-x^4\)
\(=\left(5-x^2\right)\left(5+x^2\right)\)
\(=\left(\sqrt{5}-x\right)\left(\sqrt{5}+x\right)\left(5+x^2\right)\)
b, \(x^2-6x-2=x^2-6x+9-11=\left(x-3\right)^2-\sqrt{11}^2\)
\(=\left(x-3-\sqrt{11}\right)\left(x-3+\sqrt{11}\right)\)
c,\(9x^2+6x-1=\left(3x\right)^2+2.3x+1-2=\left(3x+1\right)^2-\sqrt{2}^2\)
\(=\left(3x+1-\sqrt{2}\right)\left(3x+1+\sqrt{2}\right)\)
d,\(x^8+64=\left(x^4\right)^2+8^2+16x^4-16x^4\)
\(=\left(x^4+8\right)^2-\left(4x^2\right)^2=\left(x^4+4x^2+8\right)\left(x^4-4x^2+8\right)\)
e,\(81x^4+4=\left(9x^2\right)^2+2^2+36x^2-36x^2=\left(9x^2+2\right)^2-\left(6x\right)^2\)
\(=\left(9x^2+2-6x\right)\left(9x^2+6x+2\right)\)
g,\(x^8+x^7+1\)
\(=\left(x^8+x^7+x^6\right)+\left(x^5+x^4+x^3\right)+\left(x^2+x+1\right)-\left(x^6+x^5+x^4\right)-\left(x^3+x^2+x\right)\)
\(=x^6\left(x^2+x+1\right)+x^3\left(x^2+x+1\right)+\left(x^2+x+1\right)-x^4\left(x^2+x+1\right)-x\left(x^2+x+1\right)\)\(\left(x^2+x+1\right)\left(x^6-x^4+x^3-x+1\right)\)