Questions: Solve the equation x^3 + 2x^2 - 5x - 6 = 0 given that 2 is a zero of f(x) = x^3 + 2x^2 - 5x - 6 The solution set is

Solution Steps

We are given that \(2\) is a zero of the polynomial \(f(x) = x^3 + 2x^2 - 5x - 6\). To verify this, we substitute \(x = 2\) into the polynomial:

\[ f(2) = 2^3 + 2(2)^2 - 5(2) - 6 = 8 + 8 - 10 - 6 = 0 \]

Since \(f(2) = 0\), \(2\) is indeed a zero of the polynomial.

Since \(2\) is a zero, we can use synthetic division to divide the polynomial by \(x - 2\).

The coefficients of the polynomial \(x^3 + 2x^2 - 5x - 6\) are \(1, 2, -5, -6\).

Perform synthetic division:

\[ \begin{array}{r|rrrr} 2 & 1 & 2 & -5 & -6 \\ & & 2 & 8 & 6 \\ \hline & 1 & 4 & 3 & 0 \\ \end{array} \]

The quotient is \(x^2 + 4x + 3\) with a remainder of \(0\).

Now, we solve the quadratic equation \(x^2 + 4x + 3 = 0\).

To factor the quadratic, we look for two numbers that multiply to \(3\) and add to \(4\). These numbers are \(1\) and \(3\).

Thus, the quadratic factors as:

\[ x^2 + 4x + 3 = (x + 1)(x + 3) = 0 \]

Setting each factor equal to zero gives:

\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \] \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

Final Answer