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Answer :
We want to factor the polynomial
[tex]$$
2x^6 - 4x^5 + 7x^2 + 3.
$$[/tex]
Let's work through the procedure step by step.
1. First, consider the entire polynomial:
[tex]$$
2x^6 - 4x^5 + 7x^2 + 3.
$$[/tex]
2. Check for any common factors among all the terms. The coefficients are 2, -4, 7, and 3. Since these numbers have no common divisor besides 1 and the powers of [tex]\( x \)[/tex] are different (the terms include [tex]\( x^6 \)[/tex], [tex]\( x^5 \)[/tex], [tex]\( x^2 \)[/tex], and a constant), there is no overall factor that can be taken out.
3. Next, one might try to factor by grouping. Group the terms as follows:
[tex]$$
(2x^6 - 4x^5) + (7x^2 + 3).
$$[/tex]
In the first group, [tex]\(2x^6 - 4x^5\)[/tex], we can factor out [tex]\(2x^5\)[/tex]:
[tex]$$
2x^5(x - 2).
$$[/tex]
However, in the second group, [tex]\(7x^2 + 3\)[/tex], there is no common factor or similar factorization pattern.
4. Since no further factorization is evident in any grouping or by pulling out a common factor, we conclude that the polynomial does not factor further over the integers (or, more precisely, does not simplify into products of lower-degree polynomials with integer coefficients).
Thus, the factored form of the polynomial is the same as the original expression:
[tex]$$
2x^6 - 4x^5 + 7x^2 + 3.
$$[/tex]
This is the final answer.
[tex]$$
2x^6 - 4x^5 + 7x^2 + 3.
$$[/tex]
Let's work through the procedure step by step.
1. First, consider the entire polynomial:
[tex]$$
2x^6 - 4x^5 + 7x^2 + 3.
$$[/tex]
2. Check for any common factors among all the terms. The coefficients are 2, -4, 7, and 3. Since these numbers have no common divisor besides 1 and the powers of [tex]\( x \)[/tex] are different (the terms include [tex]\( x^6 \)[/tex], [tex]\( x^5 \)[/tex], [tex]\( x^2 \)[/tex], and a constant), there is no overall factor that can be taken out.
3. Next, one might try to factor by grouping. Group the terms as follows:
[tex]$$
(2x^6 - 4x^5) + (7x^2 + 3).
$$[/tex]
In the first group, [tex]\(2x^6 - 4x^5\)[/tex], we can factor out [tex]\(2x^5\)[/tex]:
[tex]$$
2x^5(x - 2).
$$[/tex]
However, in the second group, [tex]\(7x^2 + 3\)[/tex], there is no common factor or similar factorization pattern.
4. Since no further factorization is evident in any grouping or by pulling out a common factor, we conclude that the polynomial does not factor further over the integers (or, more precisely, does not simplify into products of lower-degree polynomials with integer coefficients).
Thus, the factored form of the polynomial is the same as the original expression:
[tex]$$
2x^6 - 4x^5 + 7x^2 + 3.
$$[/tex]
This is the final answer.
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