I mean this should be in polynomial

Question

ScanSolve AI Tutor · Accepted Answer

It appears there might have been a slight typo in the original problem. The polynomial $x^3 - 2x^2 - 5x - 6$ does not have any rational roots, meaning it cannot be factored into linear factors with rational coefficients.

Assuming the problem intended to be $x^3 - 2x^2 - 5x + 6$ (a common variation that is factorable with integer roots), here's how to factor it:

Find a root of the polynomial using the Rational Root Theorem.
Let $P(x) = x^3 - 2x^2 - 5x + 6$ .
The possible rational roots are factors of the constant term (6) divided by factors of the leading coefficient (1).
Factors of 6: $\pm 1, \pm 2, \pm 3, \pm 6$ .
Let's test $x=1$ :
$P(1) = (1)^3 - 2(1)^2 - 5(1) + 6 = 1 - 2 - 5 + 6 = -1 - 5 + 6 = -6 + 6 = 0$
Since $P(1) = 0$ , $x=1$ is a root, which means $(x-1)$ is a factor of $P(x)$ .
Use synthetic division to divide $P(x)$ by $(x-1)$ .
The coefficients of $P(x)$ are $1, -2, -5, 6$ .
$\begin{array}{c|cccc} 1 & 1 & -2 & -5 & 6 \\ & & 1 & -1 & -6 \\ \hline & 1 & -1 & -6 & 0 \\ \end{array}$
The quotient is $x^2 - x - 6$ .
Factor the resulting quadratic expression.
Now we need to factor $x^2 - x - 6$ . We look for two numbers that multiply to -6 and add to -1. These numbers are -3 and 2.
So, $x^2 - x - 6 = (x-3)(x+2)$ .
Write the fully factored polynomial.
Combine the factor from Step 1 and the factors from Step 3:
$x^3 - 2x^2 - 5x + 6 = (x-1)(x-3)(x+2)$
The factored form (assuming the typo correction) is $(x-1)(x-3)(x+2)$ .

I mean this should be in polynomial

Answer

Step-by-step solution

Find a root of the polynomial using the Rational Root Theorem.

Use synthetic division to divide $P(x)$ by $(x-1)$ .

Factor the resulting quadratic expression.

Write the fully factored polynomial.

Try one yourself

Still stuck on this one?

More Mathematics Questions

I mean this should be in polynomial

Answer

Step-by-step solution

Find a root of the polynomial using the Rational Root Theorem.

Use synthetic division to divide P(x)P(x)P(x) by (x−1)(x-1)(x−1).

Factor the resulting quadratic expression.

Write the fully factored polynomial.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Use synthetic division to divide $P(x)$ by $(x-1)$ .