Answer :
Sure, let's analyze the expression [tex]\( a^4 + a^2 + 1 \)[/tex] step by step.
### Step-by-Step Solution:
1. Understand the Expression:
The given mathematical expression is [tex]\( a^4 + a^2 + 1 \)[/tex].
2. Identify the Terms:
The expression consists of three terms:
- The first term is [tex]\( a^4 \)[/tex], where [tex]\( a \)[/tex] is raised to the power of 4.
- The second term is [tex]\( a^2 \)[/tex], where [tex]\( a \)[/tex] is raised to the power of 2.
- The third term is a constant, 1.
3. Nature of the Expression:
This is a polynomial expression in the variable [tex]\( a \)[/tex]. Specifically, it's a quartic polynomial because the highest power of [tex]\( a \)[/tex] in the expression is 4.
4. Factoring (if applicable):
Let's check if the polynomial [tex]\( a^4 + a^2 + 1 \)[/tex] can be factored. We can look for patterns or use algebraic identities. Observe that:
[tex]\[ a^4 + a^2 + 1 = a^4 + a^2 + 1^2 \][/tex]
Notice that this expression resembles a sum of cubes formulation [tex]\( x^3 + y^3 + z^3 \)[/tex] but adapted to a different form. Another method to factor would be recognizing it as [tex]\( x = a^2 \)[/tex]:
[tex]\[ a^4 + a^2 + 1 = (a^2)^2 + a^2 + 1 \][/tex]
Unfortunately, it does not factor nicely further with simple algebraic techniques.
5. Examining Polynomial Roots:
To find the roots of the polynomial expression [tex]\( a^4 + a^2 + 1 = 0 \)[/tex], we can use substitution:
Let [tex]\( b = a^2 \)[/tex]:
[tex]\[ b^2 + b + 1 = 0 \][/tex]
This is a simple quadratic equation:
- Find the discriminant of [tex]\( b^2 + b + 1 \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = 1^2 - 4(1)(1) = 1 - 4 = -3 \][/tex]
- Since the discriminant is negative, the roots of this quadratic equation are complex. Using the quadratic formula [tex]\( b = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex]:
[tex]\[ b = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm \sqrt{3}i}{2} \][/tex]
- Therefore, [tex]\( b = a^2 \)[/tex]:
[tex]\[ a^2 = \frac{-1 + \sqrt{3}i}{2} \quad \text{or} \quad a^2 = \frac{-1 - \sqrt{3}i}{2} \][/tex]
Taking square roots on both sides:
[tex]\[ a = \pm \sqrt{\frac{-1 + \sqrt{3}i}{2}} \quad \text{or} \quad a = \pm \sqrt{\frac{-1 - \sqrt{3}i}{2}} \][/tex]
Given the complexity of the roots, we usually leave the expression as is when discussing general forms or solving in non-complex scenarios.
Therefore, we conclude that the polynomial expression [tex]\( a^4 + a^2 + 1 \)[/tex] is in its simplest form and represents a quartic polynomial with no real roots, only complex ones. If specific solutions are necessary, they can be appropriately handled using advanced algebraic methods or numerical techniques.
This completes our detailed analysis and discussion of the expression [tex]\( a^4 + a^2 + 1 \)[/tex].
### Step-by-Step Solution:
1. Understand the Expression:
The given mathematical expression is [tex]\( a^4 + a^2 + 1 \)[/tex].
2. Identify the Terms:
The expression consists of three terms:
- The first term is [tex]\( a^4 \)[/tex], where [tex]\( a \)[/tex] is raised to the power of 4.
- The second term is [tex]\( a^2 \)[/tex], where [tex]\( a \)[/tex] is raised to the power of 2.
- The third term is a constant, 1.
3. Nature of the Expression:
This is a polynomial expression in the variable [tex]\( a \)[/tex]. Specifically, it's a quartic polynomial because the highest power of [tex]\( a \)[/tex] in the expression is 4.
4. Factoring (if applicable):
Let's check if the polynomial [tex]\( a^4 + a^2 + 1 \)[/tex] can be factored. We can look for patterns or use algebraic identities. Observe that:
[tex]\[ a^4 + a^2 + 1 = a^4 + a^2 + 1^2 \][/tex]
Notice that this expression resembles a sum of cubes formulation [tex]\( x^3 + y^3 + z^3 \)[/tex] but adapted to a different form. Another method to factor would be recognizing it as [tex]\( x = a^2 \)[/tex]:
[tex]\[ a^4 + a^2 + 1 = (a^2)^2 + a^2 + 1 \][/tex]
Unfortunately, it does not factor nicely further with simple algebraic techniques.
5. Examining Polynomial Roots:
To find the roots of the polynomial expression [tex]\( a^4 + a^2 + 1 = 0 \)[/tex], we can use substitution:
Let [tex]\( b = a^2 \)[/tex]:
[tex]\[ b^2 + b + 1 = 0 \][/tex]
This is a simple quadratic equation:
- Find the discriminant of [tex]\( b^2 + b + 1 \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac = 1^2 - 4(1)(1) = 1 - 4 = -3 \][/tex]
- Since the discriminant is negative, the roots of this quadratic equation are complex. Using the quadratic formula [tex]\( b = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex]:
[tex]\[ b = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm \sqrt{3}i}{2} \][/tex]
- Therefore, [tex]\( b = a^2 \)[/tex]:
[tex]\[ a^2 = \frac{-1 + \sqrt{3}i}{2} \quad \text{or} \quad a^2 = \frac{-1 - \sqrt{3}i}{2} \][/tex]
Taking square roots on both sides:
[tex]\[ a = \pm \sqrt{\frac{-1 + \sqrt{3}i}{2}} \quad \text{or} \quad a = \pm \sqrt{\frac{-1 - \sqrt{3}i}{2}} \][/tex]
Given the complexity of the roots, we usually leave the expression as is when discussing general forms or solving in non-complex scenarios.
Therefore, we conclude that the polynomial expression [tex]\( a^4 + a^2 + 1 \)[/tex] is in its simplest form and represents a quartic polynomial with no real roots, only complex ones. If specific solutions are necessary, they can be appropriately handled using advanced algebraic methods or numerical techniques.
This completes our detailed analysis and discussion of the expression [tex]\( a^4 + a^2 + 1 \)[/tex].
