Finding Zeros of Quadratic Polynomial $x^2 - 2x - 15$

To find the zeros of the quadratic polynomial $x^2 - 2x - 15$, we need to find the values of $x$ for which the polynomial equals zero. This means solving the equation:

$x^2 - 2x - 15 = 0$

Factoring the Polynomial

We can solve this quadratic equation by factoring the expression $x^2 - 2x - 15$. We look for two numbers that:

• Multiply to the constant term (-15).
• Add up to the coefficient of the $x$ term (-2).

The pairs of factors for -15 are: (1, -15), (-1, 15), (3, -5), and (-3, 5). Let's check their sums:

• $1 + (-15) = -14$
• $-1 + 15 = 14$
• $3 + (-5) = -2$
• $-3 + 5 = 2$

The pair (3, -5) adds up to -2. Therefore, we can factor the quadratic polynomial as:

$ (x + 3)(x - 5) = 0 $

Determining the Zeros

For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$:

• Set the first factor to zero:

  $x + 3 = 0$

  $x = -3$

• Set the second factor to zero:

  $x - 5 = 0$

  $x = 5$

The zeros of the quadratic polynomial $x^2 - 2x - 15$ are -3 and 5.