Question

verify that the number given alongside of the cubic polynomials below are the their heroes also verify the relationship between the zeros and the coefficient in each case 2x³+x²-5x+2;1/2,1,-2

Answer 1

p(x) = 2x³+x²-5x+2
a = 2, b = 1, c = -5, d = 2

p(½) = 2(½)³+(½)²-5(½)+2
p(½) = 2/8 + 1/4 - 5/2 + 2
p(½) = 0
therefore ½ is a zero of p(x), = alpha

p(1) = 2(1)³+(1)²-5(1)+2
p(1) = 2 + 1 - 5 + 2
p(1) = 0
therefore 1 is a zero of p(x), = beta

p(-2) = 2(-2)³+(-2)²-5(-2)+2
p(-2) = -16 + 4 + 10 + 2
p(-2) = 0
therefore -2 is a zero of p(x), = gamma

-b/a = alpha+beta+gamma
-1/2 = ½+1-2
-1/2 = -1/2

c/a = alpha×beta + beta×gamma + gamma×alpha
-5/2 = ½×1+1×-2+-2×½
-5/2 = -5/2

-d/a = alpha×beta×gamma
-2/2 = ½×1×-2
-1 = -1