Lesson 5: Using Intercept Form

Property

If the product of two numbers is zero, then one (or both) of the numbers must be zero. Using symbols,

If $AB = 0$, then either $A = 0$ or $B = 0$.

Examples

• To solve $(x - 7)(x + 2) = 0$, we set each factor to zero. This gives us $x - 7 = 0$ or $x + 2 = 0$, so the solutions are $x = 7$ and $x = -2$.
• If $y(y - 10) = 0$, then either $y = 0$ or $y - 10 = 0$. The two solutions for the equation are $y = 0$ and $y = 10$.
• Given $(2a + 1)(a - 5) = 0$, we solve $2a + 1 = 0$ to get $a = -\frac{1}{2}$, and we solve $a - 5 = 0$ to get $a = 5$.

Property

If a quadratic function has zeros at $x = p$ and $x = q$, then it can be written in intercept form as $f(x) = a(x - p)(x - q)$ where $a \neq 0$.

Examples

Property

To find zeros of a cubic polynomial $f(x) = a(x - p)(x - q)(x - r)$, set each factor equal to zero: $(x - p) = 0$, $(x - q) = 0$, and $(x - r) = 0$. For cubic polynomials not in factored form, factor completely first, then apply the zero-product property.

Examples