Question

Kiran thinks he knows one of the linear factors of P(x) = x^3 + x^2 - 17х + 15. After finding that P(3) = 0, Kiran suspects that x — 3 is a factor of P(x), so he sets up a diagram to check.

Here is the diagram he made to check his reasoning, but he set it up incorrectly. What went wrong?

Answer 1

Kiran's diagram is incorrect because he is dividing P(x) by
`(x - 3)^3` instead of (x - 3).

Kiran's diagram is incorrect because he is dividing P(x) by
`(x - 3)^3` instead of (x - 3). This is a common mistake, as it is easy to confuse the factor theorem with the polynomial long division algorithm.

When using the factor theorem, we are essentially dividing P(x) by (x - a) to check if a is a root of the polynomial. This means that we should set the remainder of the division equal to zero and solve for the coefficients of the quotient polynomial.

In Kiran's case, he is dividing P(x) by
`(x - 3)^3`. This is not a valid way to use the factor theorem, as it will always result in a remainder of zero. To correctly check if x - 3 is a factor of P(x), he should divide P(x) by (x - 3) and check if the remainder is zero.

Here is the correct method for checking if x - 3 is a factor of P(x):

`x^3` +
`x^2` - 17x + 15 | x - 3

-----------------------

`x^2` + 4x - 5

`x^2` + 4x - 5 | x - 3

-----------------------

0

As the remainder is zero, we can conclude that x - 3 is a factor of P(x).