Question

Stosując metodę wyłączania wspólnego czynnika, rozłóż wielomian W na czynniki

a) W(x)=5(x²-4)-(x-2)²

b) W(x)=x(9-x²)-(x+3)²

c) W(x)=7x(4x²+4x+1)-(4x²-1)

d) W(x)=(5-3x)(x+4)+(3x-5)(2x-3)-25+9x²

e) W(x)=3(x³-1)-2(x-1)³

f) W(x)=x⁵-3x³-4x

g) W(x)=x⁴(x-1)-4x²(x-1)+4(x-1)

Bardzo proszę o pomoc. ;-)

Answer 1

a) W(x) = 5(x² - 4) - (x - 2)² = 5(x + 2)(x - 2) - (x - 2)² = (x - 2)[5(x + 2) - (x - 2)] = (x - 2)[5x + 10 - x + 2] = (x - 2)[4x + 12] = 4(x - 2)(x + 3)

 

b)W(x) = x(9 - x²) - (x + 3)² = x(3 - x)(3 + x) - (x + 3)² = (x + 3)[x(3 - x) - (x + 3)] = (x + 3)[3x - x² - x - 3] = (x + 3)[2x - x² - 3] = -(x + 3)[x² - 2x + 3]

 

c) W(x) = 7x(4x² + 4x + 1) - (4x² - 1) = 7x(2x + 1)² - (2x + 1)(2x - 1) = (2x + 1)[7x(2x + 1) - (2x - 1)] = (2x + 1)[14x² + 7x - 2x + 1] = (2x + 1)[14x² + 5x + 1]

 

d)W(x) = (5 - 3x)(x + 4) + (3x - 5)(2x - 3) - 25 + 9x² = (5 - 3x)(x + 4) + (3x - 5)(2x - 3) - (25 - 9x²) = (5 - 3x)(x + 4) + (3x - 5)(2x - 3) - (5 - 3x)(5 + 3x) = (5 - 3x)(x + 4) - (5 - 3x)(2x - 3) - (5 - 3x)(5 + 3x) = (5 - 3x)[(x + 4) - (2x - 3) - (5 + 3x)] = (5 - 3x)[x + 4 - 2x + 3 - 5 - 3x] = (5 - 3x)[- 4x + 2] = 2(3x - 5)(2x - 1)

 

e)W(x) = 3(x³ - 1) - 2(x - 1)³ = 3(x - 1)(x² + x + 1) - 2(x - 1)³ = (x - 1)[3(x² + x + 1) - 2(x - 1)²] = (x - 1)[3x² + 3x + 3 - 2x² + 4x - 2] = (x - 1)[x² + 7x + 1]
Δ = 49 - 4 = 45
√Δ = 3√5
W(x) = (x - 1)[x² + 7x + 1] = (x - 1)(x + (7 - 3√5)/2)(x + (7 + 3√5)/2)

 

 

f)W(x) = x⁵ - 3x³ - 4x = x(x⁴ - 3x² - 4) = x(x² - 4)(x² + 1) = x(x - 2)(x + 2)(x² + 1)

 

g) W(x) = x⁴(x - 1) - 4x²(x - 1) + 4(x - 1) = (x - 1)(x⁴ - 4x² + 4) = (x - 1)(x² - 2)² = (x - 1)(x - √2)²(x + √2)²

Answer 2

Wzory skróconego mnożenia:

(a+b)²=a²+2ab+b² - kwadrat sumy;

(a-b)²=a²-2ab+b² - kwadrat różnicy;

a²-b²=(a-b)(a+b) - różnica kwadratów;

(a+b)³=a³+3a²b+3ab²+b³ - sześcian sumy;

(a-b)³=a³-3a²b+3ab²-b³ - sześcian różnicy;

a³+b³=(a+b)(a²-ab+b²) - suma sześcianów;

a³-b³=(a-b)(a²+ab+b²) - różnica sześcianów;

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a) W(x)=5(x²-4)-(x-2)²

W(x)=5(x-2)(x+2)-(x-2)²

W(x)=(x-2)[5(x+2)-(x-2)]

W(x)=(x-2)[5x+10-x+2]

W(x)=(x-2)(4x+12)

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b) W(x)=x(9-x²)-(x+3)²

W(x)=x(3-x)(3+x)-(x+3)²

W(x)=(x+3)[x(3-x)-(x+3)]

W(x)=(x+3)[3x-x²-x-3]

W(x)=(x+3)(-x²+2x-3)

W(x)=-(x+3)(x²-2x+3)

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x²-2x+3=0

Δ=b²-4ac=(-2)²-4*1*3=4-12=-8<0

Wyrażenia: x²-2x+3 nie można rozłożyć.

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c) W(x)=7x(4x²+4x+1)-(4x²-1)

W(x)=7x(2x+1)²-(2x-1)(2x+1)

W(x)=(2x+1)[7x(2x+1)-(2x-1)]

W(x)=(2x+1)[14x²+7x-2x+1]

W(x)=(2x+1)(14x²+5x+1)

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14x²+5x+1=0

Δ=b²-4ac=5²-4*14*1=25-56=-31<0

Wyrażenia: 14x²+5x+1 nie można rozłożyć.

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d) W(x)=(5-3x)(x+4)+(3x-5)(2x-3)-25+9x²

W(x)=(5-3x)(x+4)-(5-3x)(2x-3)-(5+3x)(5-3x)

W(x)=(5-3x)[(x+4)-(2x-3)-(5+3x)]

W(x)=(5-3x)[x+4-2x+3-5-3x]

W(x)=(5-3x)(2-4x)

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e) W(x)=3(x³-1)-2(x-1)³

W(x)=3(x-1)(x²+x+1)-2(x-1)³

W(x)=(x-1)[3(x²+x+1)-2(x-1)²]

W(x)=(x-1)[3x²+3x+3-2x²+4x-2]

W(x)=(x-1)(x²+7x+1)

W(x)=(x-1)(x + [7+3√5]/2)(x - [3√5-7]/2)

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x²+7x+1=0

Δ=b²-4ac=7²-4*1*1=49-4=45

√Δ=√45=3√5

x₁=[-b-√Δ]/2a=[-7-3√5]/2=-[7+3√5]/2

x₂=[-b+√Δ]/2a=[3√5-7]/2

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f) W(x)=x⁵-3x³-4x

W(x)=x⁵+x³-4x³-4x

W(x)=x³(x²+1)-4x(x²+1)

W(x)=(x³-4x)(x²+1)

W(x)=x(x²-4)(x²+1)

W(x)=x(x-2)(x+2)(x²+1)

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g) W(x)=x⁴(x-1)-4x²(x-1)+4(x-1)

W(x)=(x-1)[x⁴-4x²+4]

W(x)=(x-1)(x²-2)²

W(x)=(x-1)[(x-√2)(x+√2)]²