Question

One root of f(x) = 2x3 9x2 7x – 6 is –3. explain how to find the factors of the polynomial.

Answer 1

We have the following polynomial:


`f(x)=2x^(3)+9x^(2)+7x-6`

The problem states that one root is -3. Thus, it is true that
`(x+3)` is a factor of the polynomial. Given that this is fulfilled, it is also true that:


`f(x)=(x+3)Q(x) \therefore Q(x)=(f(x))/(x+3) \\ \\ where \ Q(x) \ has \ a \ degree \ of \ 2`

We can find Q(x) by applying Ruffini's rule, thus:


`\ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ 9 \ \ \ \ \ \ \ \ 7 \ \ \ \ \ \ -6 \\ -3 \\ \rule{50mm}{0.1mm} \\ \ {} \ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ 3 \ \ \ \ \ -2 \ \ \ \ \ \ \ \ 0`

Therefore:


`Q(x)=2x^(2)+3x-2`

The roots of this polynomial can be get as follows:


`x_(12)=(-b\pm√(b^2-4ac))/(2a) \rightarrow x_(12)=(-3\pm√(3^2-4(2)(-2)))/(2(2))\\x_(1)=(1)/(2);\ x_(2)=-2`

These are the roots along with
`-3`. Finally, the factored polynomial can be written as follows:


`f(x)=(x+3)(x+2)(2x-1)`

Answer 2

Identify the factor x + 3 from the given root –3.

Use synthetic division to divide the polynomial by x + 3.

Use the bottom row of the synthetic division as coefficients in the quadratic 2x2 + 3x – 2.

Factor the quotient to find the two other factors: x + 2 and 2x – 1.