Question

select the correct answer.

if the (x)-intercepts of a quadratic equation are ((3,0)) and ((8,0)), what are the factors of the equation?

(a) ((x - 8)) and ((x + 3))
(b) ((x - 8)) and ((x - 3))
(c) ((x + 8)) and ((x - 3))
(d) ((x + 8)) and ((x + 3))

Explanation:

🆕 New Concept Discovered: Factor Theorem for Quadratics
Connecting graph intercepts directly to algebraic factors.

Step 1: Identify the roots

The \(x\)-intercepts of a quadratic equation represent its roots (or zeros). This is where the graph crosses the \(x\)-axis and the value of the equation is zero:

$$x = 3$$

$$x = 8$$

Step 2: Apply the Factor Theorem

The Factor Theorem states that if \(x = r\) is a root of a polynomial, then \((x - r)\) is a factor of that polynomial.

To find the factors, we rewrite each root equation so that it equals zero:

• For \(x = 3\), subtract \(3\) from both sides:

$$x - 3 = 0$$

• For \(x = 8\), subtract \(8\) from both sides:

$$x - 8 = 0$$

Therefore, the factors of the quadratic equation are \((x - 3)\) and \((x - 8)\).

Answer:

\((x - 8)\) and \((x - 3)\)