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Answer :
We are given four expressions and asked to determine which one is a prime (irreducible) polynomial over the integers. A prime polynomial has no factors (other than trivial ones like constants or itself) with integer coefficients.
Let's analyze each option step by step.
1. Option A:
The polynomial is
[tex]$$
x^4 + 20x^2 - 100.
$$[/tex]
We can view this as a quadratic in [tex]$x^2$[/tex]. Let [tex]$u = x^2$[/tex], then the expression becomes
[tex]$$
u^2 + 20u - 100.
$$[/tex]
The discriminant of this quadratic is
[tex]$$
\Delta = 20^2 - 4(1)(-100) = 400 + 400 = 800.
$$[/tex]
Since [tex]$800$[/tex] is not a perfect square, there are no factors with integer coefficients. Thus, no non-trivial factorization exists over the integers, so this expression is prime.
2. Option B:
The polynomial is
[tex]$$
10x^4 - 5x^3 + 70x^2 + 3x.
$$[/tex]
Notice that each term contains a factor of [tex]$x$[/tex]. Factoring [tex]$x$[/tex] out, we have
[tex]$$
x(10x^3 - 5x^2 + 70x + 3).
$$[/tex]
Since we have factored out a non-trivial factor ([tex]$x$[/tex]), the polynomial is composite.
3. Option C:
The expression is
[tex]$$
3x^2 + 18y.
$$[/tex]
There is a common factor of [tex]$3$[/tex], so factoring it out we obtain
[tex]$$
3(x^2 + 6y).
$$[/tex]
The presence of this common factor indicates that the polynomial is not prime.
4. Option D:
The expression is
[tex]$$
x^3 - 27y^6.
$$[/tex]
Recognize that [tex]$27y^6$[/tex] can be written as [tex]$(3y^2)^3$[/tex]. Thus, the polynomial can be seen as a difference of cubes:
[tex]$$
x^3 - (3y^2)^3.
$$[/tex]
The difference of cubes factors as
[tex]$$
(x - 3y^2)\left(x^2 + 3xy^2 + 9y^4\right),
$$[/tex]
which shows that the polynomial is composite.
Since only Option A is irreducible (prime) over the integers, the correct answer is Option A.
Thus, the final answer is:
1
Let's analyze each option step by step.
1. Option A:
The polynomial is
[tex]$$
x^4 + 20x^2 - 100.
$$[/tex]
We can view this as a quadratic in [tex]$x^2$[/tex]. Let [tex]$u = x^2$[/tex], then the expression becomes
[tex]$$
u^2 + 20u - 100.
$$[/tex]
The discriminant of this quadratic is
[tex]$$
\Delta = 20^2 - 4(1)(-100) = 400 + 400 = 800.
$$[/tex]
Since [tex]$800$[/tex] is not a perfect square, there are no factors with integer coefficients. Thus, no non-trivial factorization exists over the integers, so this expression is prime.
2. Option B:
The polynomial is
[tex]$$
10x^4 - 5x^3 + 70x^2 + 3x.
$$[/tex]
Notice that each term contains a factor of [tex]$x$[/tex]. Factoring [tex]$x$[/tex] out, we have
[tex]$$
x(10x^3 - 5x^2 + 70x + 3).
$$[/tex]
Since we have factored out a non-trivial factor ([tex]$x$[/tex]), the polynomial is composite.
3. Option C:
The expression is
[tex]$$
3x^2 + 18y.
$$[/tex]
There is a common factor of [tex]$3$[/tex], so factoring it out we obtain
[tex]$$
3(x^2 + 6y).
$$[/tex]
The presence of this common factor indicates that the polynomial is not prime.
4. Option D:
The expression is
[tex]$$
x^3 - 27y^6.
$$[/tex]
Recognize that [tex]$27y^6$[/tex] can be written as [tex]$(3y^2)^3$[/tex]. Thus, the polynomial can be seen as a difference of cubes:
[tex]$$
x^3 - (3y^2)^3.
$$[/tex]
The difference of cubes factors as
[tex]$$
(x - 3y^2)\left(x^2 + 3xy^2 + 9y^4\right),
$$[/tex]
which shows that the polynomial is composite.
Since only Option A is irreducible (prime) over the integers, the correct answer is Option A.
Thus, the final answer is:
1
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