Answer :
To find the polynomial function of the lowest degree with a leading coefficient of 1 and the roots [tex]\( \sqrt{3}, -4, \)[/tex] and [tex]\( 4 \)[/tex], let's go through the process of constructing the polynomial step-by-step.
1. Identify the roots: The roots of the polynomial are [tex]\( \sqrt{3}, -4, \)[/tex] and [tex]\( 4 \)[/tex].
2. Understand the conjugate pair: Since polynomials with real coefficients require that non-real roots appear in conjugate pairs, the root [tex]\( \sqrt{3} \)[/tex] implies its conjugate [tex]\( - \sqrt{3} \)[/tex] is also a root. Since all roots are given in real numbers and not as complex ones, no need to account for complex conjugates, but it’s good to note that the structure would accommodate real pairs as well.
3. Construct factors from the roots: Each root contributes a factor to the polynomial.
- From [tex]\( \sqrt{3} \)[/tex], we get the factor [tex]\((x - \sqrt{3})\)[/tex].
- From [tex]\( -\sqrt{3} \)[/tex], we get the factor [tex]\((x + \sqrt{3})\)[/tex].
- From [tex]\(-4\)[/tex], we get the factor [tex]\((x + 4)\)[/tex].
- From [tex]\(4\)[/tex], we get the factor [tex]\((x - 4)\)[/tex].
4. Multiply the factors: To form the polynomial, we multiply these factors together.
- First, multiply the conjugate pair: [tex]\((x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3\)[/tex].
- Next, multiply the other pair: [tex]\((x + 4)(x - 4) = x^2 - 16\)[/tex].
5. Combine the results: Now multiply the results of the step above to find the complete polynomial.
- Multiply the two quadratic expressions: [tex]\((x^2 - 3)(x^2 - 16)\)[/tex].
- Expand this product:
[tex]\((x^2 - 3)(x^2 - 16) = x^2 \cdot x^2 + x^2 \cdot (-16) - 3 \cdot x^2 - 3 \cdot (-16)\)[/tex].
Simplify:
[tex]\[ x^4 - 16x^2 - 3x^2 + 48 = x^4 - 19x^2 + 48. \][/tex]
6. Conclude the polynomial: Hence, the polynomial you are seeking is:
[tex]\[ f(x) = x^4 - 19x^2 + 48. \][/tex]
So, the polynomial function of the lowest degree with a leading coefficient of 1 and the given roots is:
[tex]\[ f(x) = x^4 - 19x^2 + 48. \][/tex]
1. Identify the roots: The roots of the polynomial are [tex]\( \sqrt{3}, -4, \)[/tex] and [tex]\( 4 \)[/tex].
2. Understand the conjugate pair: Since polynomials with real coefficients require that non-real roots appear in conjugate pairs, the root [tex]\( \sqrt{3} \)[/tex] implies its conjugate [tex]\( - \sqrt{3} \)[/tex] is also a root. Since all roots are given in real numbers and not as complex ones, no need to account for complex conjugates, but it’s good to note that the structure would accommodate real pairs as well.
3. Construct factors from the roots: Each root contributes a factor to the polynomial.
- From [tex]\( \sqrt{3} \)[/tex], we get the factor [tex]\((x - \sqrt{3})\)[/tex].
- From [tex]\( -\sqrt{3} \)[/tex], we get the factor [tex]\((x + \sqrt{3})\)[/tex].
- From [tex]\(-4\)[/tex], we get the factor [tex]\((x + 4)\)[/tex].
- From [tex]\(4\)[/tex], we get the factor [tex]\((x - 4)\)[/tex].
4. Multiply the factors: To form the polynomial, we multiply these factors together.
- First, multiply the conjugate pair: [tex]\((x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3\)[/tex].
- Next, multiply the other pair: [tex]\((x + 4)(x - 4) = x^2 - 16\)[/tex].
5. Combine the results: Now multiply the results of the step above to find the complete polynomial.
- Multiply the two quadratic expressions: [tex]\((x^2 - 3)(x^2 - 16)\)[/tex].
- Expand this product:
[tex]\((x^2 - 3)(x^2 - 16) = x^2 \cdot x^2 + x^2 \cdot (-16) - 3 \cdot x^2 - 3 \cdot (-16)\)[/tex].
Simplify:
[tex]\[ x^4 - 16x^2 - 3x^2 + 48 = x^4 - 19x^2 + 48. \][/tex]
6. Conclude the polynomial: Hence, the polynomial you are seeking is:
[tex]\[ f(x) = x^4 - 19x^2 + 48. \][/tex]
So, the polynomial function of the lowest degree with a leading coefficient of 1 and the given roots is:
[tex]\[ f(x) = x^4 - 19x^2 + 48. \][/tex]
