The expression \(a^3 + 2a - 3\) cannot be factored using simple linear factors. However, we can utilize the rational root theorem to determine if it has any rational roots.
The rational root theorem states that any rational root of the polynomial equation \(a^3 + 2a - 3 = 0\) must be of the form ±(factor of the constant term) / (factor of the leading coefficient).
For the polynomial \(a^3 + 2a - 3\), the constant term is -3 and the leading coefficient is 1.
The factors of the constant term (-3) are ±1, ±3. The factors of the leading coefficient (1) are ±1.
So, the possible rational roots are \(±1, ±3\).
By evaluating the polynomial for these values of \(a\), we can check for roots:
For \(a = 1\):
\(1^3 + 2(1) - 3 = 1 + 2 - 3 = 0\)
\(a = 1\) is a root.
Now, we can perform synthetic division to factorize the expression:
\[a^3 + 2a - 3 = (a - 1)(a^2 + a + 3)\]
Therefore, the factorization of \(a^3 + 2a - 3\) is \((a - 1)(a^2 + a + 3)\). The quadratic term \(a^2 + a + 3\) cannot be further factored using real numbers.
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Answer:
The expression \(a^3 + 2a - 3\) cannot be factored using simple linear factors. However, we can utilize the rational root theorem to determine if it has any rational roots.
The rational root theorem states that any rational root of the polynomial equation \(a^3 + 2a - 3 = 0\) must be of the form ±(factor of the constant term) / (factor of the leading coefficient).
For the polynomial \(a^3 + 2a - 3\), the constant term is -3 and the leading coefficient is 1.
The factors of the constant term (-3) are ±1, ±3. The factors of the leading coefficient (1) are ±1.
So, the possible rational roots are \(±1, ±3\).
By evaluating the polynomial for these values of \(a\), we can check for roots:
For \(a = 1\):
\(1^3 + 2(1) - 3 = 1 + 2 - 3 = 0\)
\(a = 1\) is a root.
Now, we can perform synthetic division to factorize the expression:
\[a^3 + 2a - 3 = (a - 1)(a^2 + a + 3)\]
Therefore, the factorization of \(a^3 + 2a - 3\) is \((a - 1)(a^2 + a + 3)\). The quadratic term \(a^2 + a + 3\) cannot be further factored using real numbers.