Sure, let's simplify and combine like terms step-by-step for the given expression.
We start with the expression:
[tex]\[ 6x^3 + x^2 + x + x^2y + xy + y \][/tex]
1. Identify and group like terms:
- Cubic terms: [tex]\(6x^3\)[/tex]
- Quadratic terms: [tex]\(x^2\)[/tex], [tex]\(x^2y\)[/tex]
- Linear terms: [tex]\(x\)[/tex], [tex]\(xy\)[/tex], [tex]\(y\)[/tex]
2. Combine the terms by their degrees and coefficients:
- Cubic term: There is only one [tex]\(x^3\)[/tex] term, [tex]\(6x^3\)[/tex]:
[tex]\[ 6x^3 \][/tex]
- Quadratic terms: Combine [tex]\(x^2\)[/tex] and [tex]\(x^2y\)[/tex]:
[tex]\[ x^2 + x^2y \][/tex]
- Linear terms: Combine [tex]\(x\)[/tex], [tex]\(xy\)[/tex], and [tex]\(y\)[/tex]:
[tex]\[ x + xy + y \][/tex]
3. Reconstruct the simplified expression by summing all grouped terms:
Putting them back together:
[tex]\[ 6x^3 + x^2y + x^2 + xy + x + y \][/tex]
Thus, simplifying the expression [tex]\( 6 x^3 + x^2 + x + x^2 y + x y + y \)[/tex] gives us:
[tex]\[ 6x^3 + x^2y + x^2 + xy + x + y \][/tex]
