erweitern \sqrt[3]{x-5}(x^2+6x+5)(x-3)^5

Lösung

erweitern

3 x - 5 (x^{2} + 6 x + 5) (x - 3)^{5}

x^{7} 3 x - 5 - 9 x^{6} 3 x - 5 + 5 x^{5} 3 x - 5 + 195 x^{4} 3 x - 5 - 765 x^{3} 3 x - 5 + 837 x^{2} 3 x - 5 + 567 x 3 x - 5 - 1215 3 x - 5

Schritte zur Lösung

3 x - 5 (x^{2} + 6 x + 5) (x - 3)^{5}

= (x - 3)^{5} 3 x - 5 (x^{2} + 6 x + 5)

Wende das Distributivgesetz an:

m (a + b + c) = ma + mb + m c

(x - 3)^{5} 3 x - 5 (x^{2} + 6 x + 5) = (x - 3)^{5} 3 x - 5 x^{2} + (x - 3)^{5} 3 x - 5 \cdot 6 x + (x - 3)^{5} 3 x - 5 \cdot 5

= (x - 3)^{5} 3 x - 5 x^{2} + (x - 3)^{5} 3 x - 5 \cdot 6 x + (x - 3)^{5} 3 x - 5 \cdot 5

= x^{2} (x - 3)^{5} 3 x - 5 + 6 x (x - 3)^{5} 3 x - 5 + 5 (x - 3)^{5} 3 x - 5

Schreibe

x^{2} (x - 3)^{5} 3 x - 5

um:

x^{7} 3 x - 5 - 15 x^{6} 3 x - 5 + 90 x^{5} 3 x - 5 - 270 x^{4} 3 x - 5 + 405 x^{3} 3 x - 5 - 243 x^{2} 3 x - 5

= x^{7} 3 x - 5 - 15 x^{6} 3 x - 5 + 90 x^{5} 3 x - 5 - 270 x^{4} 3 x - 5 + 405 x^{3} 3 x - 5 - 243 x^{2} 3 x - 5 + 6 x (x - 3)^{5} 3 x - 5 + 5 (x - 3)^{5} 3 x - 5

Schreibe

6 x (x - 3)^{5} 3 x - 5

um:

6 x^{6} 3 x - 5 - 90 x^{5} 3 x - 5 + 540 x^{4} 3 x - 5 - 1620 x^{3} 3 x - 5 + 2430 x^{2} 3 x - 5 - 1458 x 3 x - 5

= x^{7} 3 x - 5 - 15 x^{6} 3 x - 5 + 90 x^{5} 3 x - 5 - 270 x^{4} 3 x - 5 + 405 x^{3} 3 x - 5 - 243 x^{2} 3 x - 5 + 6 x^{6} 3 x - 5 - 90 x^{5} 3 x - 5 + 540 x^{4} 3 x - 5 - 1620 x^{3} 3 x - 5 + 2430 x^{2} 3 x - 5 - 1458 x 3 x - 5 + 5 (x - 3)^{5} 3 x - 5

Schreibe

5 (x - 3)^{5} 3 x - 5

um:

5 x^{5} 3 x - 5 - 75 x^{4} 3 x - 5 + 450 x^{3} 3 x - 5 - 1350 x^{2} 3 x - 5 + 2025 x 3 x - 5 - 1215 3 x - 5

= x^{7} 3 x - 5 - 15 x^{6} 3 x - 5 + 90 x^{5} 3 x - 5 - 270 x^{4} 3 x - 5 + 405 x^{3} 3 x - 5 - 243 x^{2} 3 x - 5 + 6 x^{6} 3 x - 5 - 90 x^{5} 3 x - 5 + 540 x^{4} 3 x - 5 - 1620 x^{3} 3 x - 5 + 2430 x^{2} 3 x - 5 - 1458 x 3 x - 5 + 5 x^{5} 3 x - 5 - 75 x^{4} 3 x - 5 + 450 x^{3} 3 x - 5 - 1350 x^{2} 3 x - 5 + 2025 x 3 x - 5 - 1215 3 x - 5

Vereinfache

x^{7} 3 x - 5 - 15 x^{6} 3 x - 5 + 90 x^{5} 3 x - 5 - 270 x^{4} 3 x - 5 + 405 x^{3} 3 x - 5 - 243 x^{2} 3 x - 5 + 6 x^{6} 3 x - 5 - 90 x^{5} 3 x - 5 + 540 x^{4} 3 x - 5 - 1620 x^{3} 3 x - 5 + 2430 x^{2} 3 x - 5 - 1458 x 3 x - 5 + 5 x^{5} 3 x - 5 - 75 x^{4} 3 x - 5 + 450 x^{3} 3 x - 5 - 1350 x^{2} 3 x - 5 + 2025 x 3 x - 5 - 1215 3 x - 5 : x^{7} 3 x - 5 - 9 x^{6} 3 x - 5 + 5 x^{5} 3 x - 5 + 195 x^{4} 3 x - 5 - 765 x^{3} 3 x - 5 + 837 x^{2} 3 x - 5 + 567 x 3 x - 5 - 1215 3 x - 5

= x^{7} 3 x - 5 - 9 x^{6} 3 x - 5 + 5 x^{5} 3 x - 5 + 195 x^{4} 3 x - 5 - 765 x^{3} 3 x - 5 + 837 x^{2} 3 x - 5 + 567 x 3 x - 5 - 1215 3 x - 5

e x p an d 5 (2 + 1)

e x p an d \frac{( a + b ) x + ( a - b ) ∣ x ∣}{2 x}

s im pl i f y \frac{1}{2} (- \frac{1}{4} cos (4 θ) - \frac{1}{2} cos (2 θ))

e x p an d (- 1, cos (t)) \cdot (2 sin (t), 3 π - 3 t)

e x p an d \frac{( 3 x + 2 ) ^{2}}{4 ( x + 2 ) ^{2}}