Question

Daje dużo punktów, ale oczekuje rzetelnych odpowiedzi w innym przypadku zgłaszam spam. Ma być rozwiązane krok po kroku.
1. Stosując metodę wyłączania wspólnego czynnika, rozłóż wielomian W na czynniki.
a) W(x)= x²(x-3)-4(x-3)
b) W(x)= x²(x-1)-x+1
c) W(x)= x²(2x+5)-2x-5
d) W(x)= x²(x+2)+x+2
e) W(x)= (4x²-9)(x+1)+3(2x+3)
f) W(x)= 5(x²-4)-(x-2)²
g) W(x)= x(9-x²)-(x+3)²
h) W(x)= 7x(4x²+4x+1)-(4x²-1)
i) W(x)= (5-3x)(x+4)+(3x-5)(2x-3)-25+9x²
j) W(x)= 3(x³-1)-2(x-1)³
k) W(x)= x⁵-3x³-4x
l) W(x)= x⁴(x-1)-4x²(x-1)+4(x-1)

2. Stosując wzory skróconego mnożenia, rozłóż wielomian W na czynniki, gdy:
a) W(x)= (x+2)³+27
b) W(x)= 125-(x+3)³
c) W(x)= (x+1)²-(2x+2)²
d) W(x)=(2m+1)²-(m+3)²
e) W(x)= a³-3
f) W(x)=1-6x³
g) W(x)= 0,25y²-1,44y⁴
h) W(x)= x⁴-10x²+25

Answer 1

1. Stosując metodę wyłączania wspólnego czynnika, rozłóż wielomian W na czynniki.
a) W(x) = x²(x - 3) - 4(x - 3) = (x - 3)(x² - 4) = (x - 3)(x - 2)(x + 2)

b) W(x) = x²(x - 1) - x + 1 = x²(x - 1) - (x - 1) = (x - 1)(x² - 1) = (x - 1)(x - 1)(x + 1) = (x - 1)²(x + 1)

c) W(x) = x²(2x + 5) - 2x - 5 = x²(2x+5) - (2x + 5) = (2x + 5)(x² - 1) = (2x + 5)(x + 1)(x - 1)

d) W(x) = x²(x + 2) + x + 2 = (x + 2)(x² + 1)

e) W(x) = (4x² - 9)(x + 1) + 3(2x + 3) = (2x - 3)(2x + 3)(x + 1) + 3(2x + 3) = (2x + 3)[(2x - 3)(x + 1) + 3] = (2x + 3)[2x² - 3x + 2x - 3 + 3] = (2x + 3)[2x² - x] = x(2x + 3)(2x - 1)

f) 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)

g) 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]

h) 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]

i) 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)

j) 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)

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

l) 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

2. Stosując wzory skróconego mnożenia, rozłóż wielomian W na czynniki, gdy:
a) W(x) = (x + 2)³ + 27 = [(x + 2) + 3][(x + 2)² - 3(x + 2) + 9] = (x + 5)[x² + 4x + 4 - 3x - 6 + 9] = (x + 5)[x² + x + 7]

b) W(x) = 125 - (x + 3)³ = [5 - (x + 3)][25 + 5(x + 3) + (x + 3)²] = (2 - x)(25 + 5x + 15 + x² + 6x + 9) = (2 - x)(x² + 11x + 49)

c) W(x) = (x + 1)² - (2x + 2)² = (x + 1)² - 4(x + 1)² = -3(x + 1)²

d) W(x) = (2m + 1)² - (m + 3)² = [(2m + 1) - (m + 3)][(2m + 1) + (m + 3)] = (m - 2)(3m + 4)

e) W(x) = a³ - 3 = (a - ³√3)(a² + a³√3 + ³√9)

f) W(x) = 1 - 6x³ = (1 - ³√6x)(1 + ³√6x + ³√36x²)

g) W(x) = 0,25y² - 1,44y⁴ = y²(0,5 + 1,2y)(0,5 - 1,2y)

h) W(x) = x⁴ - 10x² + 25 = (x² - 5) = (x - √5)(x + √5)

Answer 2

Zadanie 1.

a) W(x) = x²(x - 3) - 4(x - 3) = (x - 3)(x² - 4) = (x - 3)(x - 2)(x + 2)
b) W(x) = x²(x - 1) - x + 1 = x²(x - 1) - (x - 1) = (x - 1)(x² - 1) = (x - 1)(x - 1)(x + 1) = (x - 1)²(x + 1)
c) W(x) = x²(2x + 5) - 2x - 5 = x²(2x+5) - (2x + 5) = (2x + 5)(x² - 1) = (2x + 5)(x + 1)(x - 1)
d) W(x) = x²(x + 2) + x + 2 = (x + 2)(x² + 1)
e) W(x) = (4x² - 9)(x + 1) + 3(2x + 3) = (2x - 3)(2x + 3)(x + 1) + 3(2x + 3) = (2x + 3)[(2x - 3)(x + 1) + 3] = (2x + 3)[2x² - 3x + 2x - 3 + 3] = (2x + 3)[2x² - x] = x(2x + 3)(2x - 1)
f) 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)
g) 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]
h) 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]
i) 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)
j) 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)
k) W(x) = x⁵ - 3x³ - 4x = x(x⁴ - 3x² - 4) = x(x² - 4)(x² + 1) = x(x - 2)(x + 2)(x² + 1)
l) 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

Zadanie 2.

a) W(x) = (x + 2)³ + 27 = [(x + 2) + 3][(x + 2)² - 3(x + 2) + 9] = (x + 5)[x² + 4x + 4 - 3x - 6 + 9] = (x + 5)[x² + x + 7]
b) W(x) = 125 - (x + 3)³ = [5 - (x + 3)][25 + 5(x + 3) + (x + 3)²] = (2 - x)(25 + 5x + 15 + x² + 6x + 9) = (2 - x)(x² + 11x + 49)
c) W(x) = (x + 1)² - (2x + 2)² = (x + 1)² - 4(x + 1)² = -3(x + 1)²
d) W(x) = (2m + 1)² - (m + 3)² = [(2m + 1) - (m + 3)][(2m + 1) + (m + 3)] = (m - 2)(3m + 4)
e) W(x) = a³ - 3 = (a - ³√3)(a² + a³√3 + ³√9)
f) W(x) = 1 - 6x³ = (1 - ³√6x)(1 + ³√6x + ³√36x²)
g) W(x) = 0,25y² - 1,44y⁴ = y²(0,5 + 1,2y)(0,5 - 1,2y)
h) W(x) = x⁴ - 10x² + 25 = (x² - 5) = (x - √5)(x + √5)