Sure! To find the quadratic polynomial whose roots (zeroes) have a sum of 3 and a product of 5, we can use the properties of quadratic equations. Let's break this down step-by-step:
1. Understanding the problem: We need to find a quadratic polynomial [tex]\( f(x) = ax^2 + bx + c \)[/tex], where the sum of its roots is 3 and the product of its roots is 5.
2. Using the sum and product relationships:
- For a quadratic polynomial of the form [tex]\( ax^2 + bx + c \)[/tex], the sum of the roots ([tex]\( \alpha + \beta \)[/tex]) is given by [tex]\( -b/a \)[/tex].
- The product of the roots ([tex]\( \alpha \beta \)[/tex]) is given by [tex]\( c/a \)[/tex].
3. Choosing a:
- It's convenient to choose [tex]\( a = 1 \)[/tex] to simplify the calculations. This leads to [tex]\( x^2 + bx + c \)[/tex].
4. Calculating coefficients [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
- Given the sum of the roots is 3, we know that [tex]\( -b/a = 3 \)[/tex]. With [tex]\( a = 1 \)[/tex], this simplifies to [tex]\( -b = 3 \)[/tex] or [tex]\( b = -3 \)[/tex].
- Given the product of the roots is 5, we know that [tex]\( c/a = 5 \)[/tex]. With [tex]\( a = 1 \)[/tex], this simplifies to [tex]\( c = 5 \)[/tex].
5. Formulating the quadratic polynomial:
- Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the standard form [tex]\( ax^2 + bx + c \)[/tex].
- Therefore, the polynomial is [tex]\( 1x^2 + (-3)x + 5 \)[/tex].
6. Final answer:
- The quadratic polynomial with a sum of roots 3 and a product of roots 5 is [tex]\( x^2 - 3x + 5 \)[/tex].
Thus, the quadratic polynomial is:
[tex]\[ x^2 - 3x + 5 \][/tex]
