Question

For what value of k, the expression x3 + kx2 – 7x + 6 can be resolved into three linear factors?

Answer 1

We have to find the value of k for which the expression x³ + kx² - 7x + 6 can be resolved into three linear factor.

solution : here polynomial is x³ + kx² - 7x + 6 can be resolved into three linear factors.

it is possible only when all the three roots of polynomial is real numbers.

let a , b and c are three roots of polynomial.

so, product of roots = abc = -constant/coefficient of x³

⇒abc = -6 ...(1)

sum of products of two roots = ab + bc + ca = coefficient of x/coefficient of x³

⇒ab + bc + ca = -7 ...(2)

now if assume a = 1, b = 2 and c = -3

then, 1 × 2 × -3 = -6 eq (1) satisfied

1 × 2 + 2 × -3 + -3 × 1 = -7 eq (2) satisfied

therefore roots of polynomial are 1, 2 and -3

now sum of roots = - coefficient of x²/coefficient of x³

⇒a + b + c = -k

⇒1 + 2 - 3 = -k

⇒k = 0

Therefore the value of k is zero.