Question

Wyznacz punkty wspolne wykresu wielomianu w i prostej l.

a) w(x)=1/3x^3+x+2, l:y=4x+2

b)w(x)=4x^3-x^2-5x+3, l:y=3x+1

c)w(x)=x^5-4x^4+3x^3+x^2,l:y=4x-3

Answer 1

a)

1/3x³+x+2=4x+2

1/3x³-3x=0

1/3x(x²-9)=0

1/3x(x+3)(x-3)=0

x=0  v x=-3 v x=3

y=2 v y=-10  y=14

A(-3,-10), B(3,14), C(0,2)

b)

4x³-x²-5x+3=3x+1

4x³-x²-8x+2=0

x²(4x-1)-2(4x-1)=0

(4x-1)(x²-2)=0

(4x-1)(x+√2)(x-√2)=0

x=1/4 v x=-√2 v x=√2

y=7/4 v y=1-3√2 v y=1+3√2

A=(1/4,7/4), B=(-√2,, 1-3√2), C=(√2, 1+3√2)

c)

x^5-4x^4+3x³+x²=4x-3

x^5-4x^4+3x³+x²-4x+3=0

x⁴(x-1)-3x³(x-1)+x(x-1)-3(x-1)=0

(x-1)(x⁴-3x³+x-3)=0

(x-1)*[x³(x-3)+(x-3)]=0

(x-1)(x-3)(x³+1)=0

x=-1  v x=1  v x=3

y=-7 v y=1  v y=9

A=(-1,-7), B=(1, 1), C=(3, 9).