Certainly! Let's explore the polynomial [tex]\( p(x) = 3x^2 - 12x + 15 \)[/tex] and find its roots, denoted as [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex]. We need to determine the values of [tex]\(\alpha + \beta\)[/tex] and [tex]\(\alpha \beta\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients of the polynomial:
The polynomial takes the general form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = -12 \)[/tex]
- [tex]\( c = 15 \)[/tex]
2. Sum of the roots:
According to the properties of a quadratic polynomial [tex]\( ax^2 + bx + c \)[/tex], the sum of the roots ([tex]\(\alpha + \beta\)[/tex]) is given by:
[tex]\[
\alpha + \beta = -\frac{b}{a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
\alpha + \beta = -\frac{-12}{3} = \frac{12}{3} = 4.0
\][/tex]
3. Product of the roots:
Similarly, the product of the roots ([tex]\(\alpha \beta\)[/tex]) is given by:
[tex]\[
\alpha \beta = \frac{c}{a}
\][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[
\alpha \beta = \frac{15}{3} = 5.0
\][/tex]
### Conclusion:
- The sum of the roots [tex]\( \alpha + \beta \)[/tex] is [tex]\( 4.0 \)[/tex].
- The product of the roots [tex]\( \alpha \beta \)[/tex] is [tex]\( 5.0 \)[/tex].
Thus, the values are:
[tex]\[
\alpha + \beta = 4.0 \quad \text{and} \quad \alpha \beta = 5.0
\][/tex]
