verify that the number given alongside of the cubic polynomial below are the zeros. Also, verify the relationship between the zeroes and the coefficients in each case b)2x^3-5x^2+x+2. ; 1,2 and -1/2
Answers
As We know that we have some numbers and we're going to check that whether they are zeroes or not of this Polynomial 2x³–5x²+x+2.
- p(x) = 2x³-5x²+x+2
- zeroes for Polynomial = 1,2, -1/2
→ p(1) = 2(1)³-5(1)²+(1)+2
→ 2-5+1+2
→ 0
→ p(2) = 2(2)³-5(2)²+(2)+2
→ 16-20+2+2
→ 0
→ p(-½) = 2(-½)³-5(-½)²+(-½)+2
→ –¼ - 5/4 - ½ + 2
→ 0
This is proved that 1, 2, –½ are the zeroes of the Polynomial.
Now, comparing the given polynomial with general expression, we get;
→ ax³+bx³+cx+d
→ 2x³–5x²+x+2
→ a=2, b=-5, c=1, d =2
As we know, if α, β, γ are the zeroes of the cubic polynomial ax³+bx²+cx+d.
→ α +β+γ = –b/a
→ 1+2+(-½) = 5/2
→ 5/2 = –b/a
→ αβ+βγ+γα = c/a
→ (1×2)+(2 ×-½)+(-½×1) = ½
→ ½ = c/a
→ αβγ = – d/a
→ 1×2×–½ = –1
→ –1 = –d/a
Therefore, Putting the values of zeroes of the polynomial.
Hence, the relationship b/w the zeroes and the coefficients are satisfactory.