Given `g\left(x\right)=x^{2}-3x-10`, which statement is true?

Question

The zeros are 5 and -2 because the factors are (x-5) and (x+2)
The zeros are 2 and -5 because the factors are (x-2) and (x+5)
The zeros are 2 and 5 because the factors are (x-2) and (x-5)

Answer 1

To find the zeros of the function \( g(x) = x^{2} - 3x - 10 \), we can factor the quadratic expression.

We look for two numbers that multiply to \(-10\) (the constant term) and add up to \(-3\) (the coefficient of \(x\)).

The two numbers that satisfy these conditions are \(-5\) and \(2\), since:

\[ -5 \times 2 = -10 \quad \text{and} \quad -5 + 2 = -3 \]

Thus, we can factor \( g(x) \) as follows:

\[ g(x) = (x - 5)(x + 2) \]

Now, the zeros of the function occur when \( g(x) = 0 \), which gives us:

\[ (x - 5)(x + 2) = 0 \]

This leads to the following solutions:

\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

So, the zeros are \(5\) and \(-2\), and the factors are \((x - 5)\) and \((x + 2)\).

Now looking at your provided statements, the correct statement is:

The zeros are 5 and -2 because the factors are (x-5) and (x+2).