Answer :
Sure! To write a quadratic polynomial given its zeros, we will follow these steps:
1. Identify the Zeros:
- The first zero is [tex]\( -2 \sqrt{3} \)[/tex].
- The second zero is [tex]\( \frac{\sqrt{5}}{3} \)[/tex].
2. Use the General Form:
The general form of a quadratic polynomial given its zeros [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] is:
[tex]\[ P(x) = a(x - \alpha)(x - \beta) \][/tex]
For simplicity, we can take the leading coefficient [tex]\( a \)[/tex] to be 1.
3. Write the Polynomial Using the Zeros:
Substituting [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
[tex]\[ P(x) = (x + 2 \sqrt{3})(x - \frac{\sqrt{5}}{3}) \][/tex]
4. Expand the Polynomial:
To expand, we use the distributive property:
[tex]\[ P(x) = x^2 - \frac{\sqrt{5}}{3} x + 2\sqrt{3} x - \frac{2 \sqrt{3} \cdot \sqrt{5}}{3} \][/tex]
Simplifying the product term:
[tex]\[ - \frac{2 \sqrt{3} \cdot \sqrt{5}}{3} = - \frac{2 \sqrt{15}}{3} \][/tex]
5. Combine Like Terms:
Group and combine the coefficients of [tex]\( x \)[/tex]:
[tex]\[ P(x) = x^2 + (2 \sqrt{3} - \frac{\sqrt{5}}{3}) x - \frac{2 \sqrt{15}}{3} \][/tex]
6. Sum and Product of Zeros:
To express the polynomial in the standard form [tex]\( x^2 - (sum \, of \, zeros)x + (product \, of \, zeros) \)[/tex]:
- Sum of zeros:
[tex]\[ \alpha + \beta = -2 \sqrt{3} + \frac{\sqrt{5}}{3} \][/tex]
Which evaluates to approximately [tex]\( -2.7187456226378246 \)[/tex].
- Product of zeros:
[tex]\[ \alpha \cdot \beta = (-2 \sqrt{3}) \cdot \left( \frac{\sqrt{5}}{3} \right) \][/tex]
Which evaluates to approximately [tex]\( -2.581988897471611 \)[/tex].
7. Form the Polynomial:
The quadratic polynomial is:
[tex]\[ P(x) = x^2 - (sum \, of \, zeros)x + (product \, of \, zeros) \][/tex]
Substituting the obtained values:
[tex]\[ P(x) = x^2 - (-2.7187456226378246)x + (-2.581988897471611) \][/tex]
Simplifying the signs for clarity:
[tex]\[ P(x) = x^2 + 2.7187456226378246 x - 2.581988897471611 \][/tex]
So, the quadratic polynomial whose zeros are [tex]\( -2 \sqrt{3} \)[/tex] and [tex]\( \frac{\sqrt{5}}{3} \)[/tex] is:
[tex]\[ P(x) = x^2 + 2.7187456226378246 x - 2.581988897471611 \][/tex]
1. Identify the Zeros:
- The first zero is [tex]\( -2 \sqrt{3} \)[/tex].
- The second zero is [tex]\( \frac{\sqrt{5}}{3} \)[/tex].
2. Use the General Form:
The general form of a quadratic polynomial given its zeros [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] is:
[tex]\[ P(x) = a(x - \alpha)(x - \beta) \][/tex]
For simplicity, we can take the leading coefficient [tex]\( a \)[/tex] to be 1.
3. Write the Polynomial Using the Zeros:
Substituting [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:
[tex]\[ P(x) = (x + 2 \sqrt{3})(x - \frac{\sqrt{5}}{3}) \][/tex]
4. Expand the Polynomial:
To expand, we use the distributive property:
[tex]\[ P(x) = x^2 - \frac{\sqrt{5}}{3} x + 2\sqrt{3} x - \frac{2 \sqrt{3} \cdot \sqrt{5}}{3} \][/tex]
Simplifying the product term:
[tex]\[ - \frac{2 \sqrt{3} \cdot \sqrt{5}}{3} = - \frac{2 \sqrt{15}}{3} \][/tex]
5. Combine Like Terms:
Group and combine the coefficients of [tex]\( x \)[/tex]:
[tex]\[ P(x) = x^2 + (2 \sqrt{3} - \frac{\sqrt{5}}{3}) x - \frac{2 \sqrt{15}}{3} \][/tex]
6. Sum and Product of Zeros:
To express the polynomial in the standard form [tex]\( x^2 - (sum \, of \, zeros)x + (product \, of \, zeros) \)[/tex]:
- Sum of zeros:
[tex]\[ \alpha + \beta = -2 \sqrt{3} + \frac{\sqrt{5}}{3} \][/tex]
Which evaluates to approximately [tex]\( -2.7187456226378246 \)[/tex].
- Product of zeros:
[tex]\[ \alpha \cdot \beta = (-2 \sqrt{3}) \cdot \left( \frac{\sqrt{5}}{3} \right) \][/tex]
Which evaluates to approximately [tex]\( -2.581988897471611 \)[/tex].
7. Form the Polynomial:
The quadratic polynomial is:
[tex]\[ P(x) = x^2 - (sum \, of \, zeros)x + (product \, of \, zeros) \][/tex]
Substituting the obtained values:
[tex]\[ P(x) = x^2 - (-2.7187456226378246)x + (-2.581988897471611) \][/tex]
Simplifying the signs for clarity:
[tex]\[ P(x) = x^2 + 2.7187456226378246 x - 2.581988897471611 \][/tex]
So, the quadratic polynomial whose zeros are [tex]\( -2 \sqrt{3} \)[/tex] and [tex]\( \frac{\sqrt{5}}{3} \)[/tex] is:
[tex]\[ P(x) = x^2 + 2.7187456226378246 x - 2.581988897471611 \][/tex]
