To solve the problem of determining the quadratic polynomial given the sum and product of its zeros, follow these steps:
1. Understand the Relationship:
- For a quadratic polynomial of the form [tex]\( ax^2 + bx + c \)[/tex], the sum of the zeros (roots) is given by [tex]\( -\frac{b}{a} \)[/tex].
- The product of the zeros is given by [tex]\( \frac{c}{a} \)[/tex].
2. Given Information:
- The sum of the zeros is 3.
- The product of the zeros is -10.
3. Assume Standard Form (a = 1):
- Assuming [tex]\( a = 1 \)[/tex] for simplicity, the standard form of the quadratic polynomial becomes [tex]\( x^2 + bx + c \)[/tex].
4. Determine the Coefficients [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
- The sum of the zeros is [tex]\(-\frac{b}{1}\)[/tex], which is equal to 3. Therefore, [tex]\( -b = 3 \)[/tex] which implies [tex]\( b = -3 \)[/tex].
- The product of the zeros is [tex]\(\frac{c}{1}\)[/tex], which is equal to -10. Therefore, [tex]\( c = -10 \)[/tex].
5. Form the Quadratic Polynomial:
- Substituting [tex]\( b \)[/tex] and [tex]\( c \)[/tex] into the standard form [tex]\( x^2 + bx + c \)[/tex], we get:
[tex]\[ x^2 - 3x - 10 \][/tex]
Thus, the quadratic polynomial with a sum of zeros equal to 3 and a product of zeros equal to -10 is:
[tex]\[ x^2 - 3x - 10 \][/tex]
Given the options, since the correct form of the polynomial is not inclusive of your provided option, ensure you relate properly if there’s a typo or error in the question choices. However, based on the above calculated polynomial, the polynomial is actually [tex]\( x^2 - 3x - 10 \)[/tex].
