13/08/2023 08:51:16

Phân tích đa thức thành nhân tử: 1) (x^2 + 8x + 12)(x^2 + 12x + 32) + 16; 2) (x^2 + 6x + 8)(x^2 + 8x + 15) - 24; 3) (x^2 + 4x + 3)(x^2 + 6x + 8) - 24

1, (x^2 + 8x + 12)(x^2 + 12x + 32) + 16
2, (x^2 + 6x + 8)(x^2 + 8x + 15) - 24
3,(x^2 + 4x + 3)(x^2 + 6x + 8) - 24

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1. To simplify the expression (x^2 + 8x + 12)(x^2 + 12x + 32) + 16, we can expand the two binomials and then combine like terms:

(x^2 + 8x + 12)(x^2 + 12x + 32) + 16
= x^2(x^2 + 12x + 32) + 8x(x^2 + 12x + 32) + 12(x^2 + 12x + 32) + 16
= x^4 + 12x^3 + 32x^2 + 8x^3 + 96x^2 + 256x + 12x^2 + 144x + 384 + 16
= x^4 + 20x^3 + 140x^2 + 400x + 400

Therefore, the simplified expression is x^4 + 20x^3 + 140x^2 + 400x + 400.

2. To simplify the expression (x^2 + 6x + 8)(x^2 + 8x + 15) - 24, we can expand the two binomials and then combine like terms:

(x^2 + 6x + 8)(x^2 + 8x + 15) - 24
= x^2(x^2 + 8x + 15) + 6x(x^2 + 8x + 15) + 8(x^2 + 8x + 15) - 24
= x^4 + 8x^3 + 15x^2 + 6x^3 + 48x^2 + 90x + 8x^2 + 64x + 120 - 24
= x^4 + 14x^3 + 71x^2 + 154x + 96

Therefore, the simplified expression is x^4 + 14x^3 + 71x^2 + 154x + 96.

3. To simplify the expression (x^2 + 4x + 3)(x^2 + 6x + 8) - 24, we can expand the two binomials and then combine like terms:

(x^2 + 4x + 3)(x^2 + 6x + 8) - 24
= x^2(x^2 + 6x + 8) + 4x(x^2 + 6x + 8) + 3(x^2 + 6x + 8) - 24
= x^4 + 6x^3 + 8x^2 + 4x^3 + 24x^2 + 32x + 3x^2 + 18x + 24 - 24
= x^4 + 10x^3 + 35x^2 + 50x

Therefore, the simplified expression is x^4 + 10x^3 + 35x^2 + 50x.

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Thưởng th.10.2024

Xếp hạng

1, (x^2+8x+12)(x^2+12x+32)+16
= (x^2+2x+6x+12)(x^2+4x+8x+32)+16
=[x(x+2)+6(x+2)][x(x+4)+8(x+4)]+16
=(x+2)(x+4)(x+6)(x+8)+16
=[(x+2)(x+8)][(x+4)(x+6)]+16
=(x^2+10x+16)(x^2+10x+24)+16
Đặt a=x^2+10x+20
khi đó ta có
(x^2+10x+16)(x^2+10x+24)+16
= (a+4)(a-4)+16
=a^2-16+16
=a^2
=(x^2+10x+20)^2

2, (x^2 + 6x + 8)(x^2 + 8x + 15) - 24
= (x^2 +2x+4x+ 8)(x^2 +3x+5x + 15) - 24
= [x(x+2)+4(x+2)][x(x+3)+5(x+3)]-24
=(x+2)(x+3)(x+4)(x+5)-24
=[(x+2)(x+5)] [(x+3)(x+4)]-24
= (x^2+7x+10)(x^2+7x+12)-24
Đặt a=x^2+7x+11 ta có
(x^2+7x+10)(x^2+7x+12)-24
= (a+1)(a-1)-24
= a^2-1-24
= a^2-25
=(a+5)(a-5)
=(x^2+7x+11+5)(x^2+7x+11-5)
= (x^2+7x+16)(x^2+7x+6)

3, (x^2 + 4x + 3)(x^2 + 6x + 8) - 24
= (x^2 + 1x +3x+ 3)(x^2 + 2x +4x + 8) - 24
= [x(x+1)+3(x+1)] [x(x+2)+4(x+2)] -24
= (x+1)(x+2)(x+3)(x+4)-24
= [(x+1)(x+4)] [(x+2)(x+3)] -24
= (x^2+5x+4)(x^2+5x+6)-24
Đặt a=x^2+5x+5 ta có
(x^2+5x+4)(x^2+5x+6)-24
= (a+1)(a-1)-24
= a^2-1-24
= a^2-25
= (a+5)(a-5)
= (x^2+5x+5+5)(x^2+5x+5-5)
= x(x^2+5x+10)(x+5)

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Phân tích đa thức thành nhân tử: 1) (x^2 + 8x + 12)(x^2 + 12x + 32) + 16; 2) (x^2 + 6x + 8)(x^2 + 8x + 15) - 24; 3) (x^2 + 4x + 3)(x^2 + 6x + 8) - 24

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