Simplify the expression. 3 x 2 + 5 x 3 − 2 x + 4 x 2 − x 3 + 8 x + 10 3x^2+5x^3-2x+4x^2-x^3+8x+10

Here we're asked to simplify the expression three x squared plus five x cubed minus two x plus four x squared minus x cubed plus eight x plus 10. So this is a rather long expression with a lot of terms. So we need to stay super organized here and follow our steps, starting with distributing any constants or variables into parentheses. Now there are no parentheses here and nothing that needs to get distributed, so I can move on to step two, identifying my like terms. Now, this is super important here because there are a lot of terms that have x, but not all of those x's are raised to the same exponent. So we need to pay close attention to which terms are actually like. They have the same variable raised to the same exponent. So looking at my first term here, I have three x squared. And continuing on in my expression, my next term is five x cubed. Even though that's the same variable, it is not raised to the same exponent. So it is not a like term. Continuing on that negative two x, again, same variable, but not raised to the same exponent. So not a like term. Then I get to this four x square that is the same variable x raised to the same exponent of two. So that is a like term. Now negative x cubed, eight x, and 10, those are not like terms to my three x squared. So moving on to this five x cubed. So I'm looking for that variable x, but this time it cubed for any other like terms. I see one other instance of this. I have negative x cubed. So that is those are two like terms. Then I have negative two x and positive eight x. Same variable raised to the same exponent. Then my final remaining term here is this positive 10. It's just a constant. There are no other constants, so this doesn't have any like terms. So I've identified my like terms. Now I want to go ahead and group them by writing them next to each other. So starting with my X squared terms, so I have three X squared and four X squared. So writing those next to each other, three X squared plus four X squared. Then moving on to my X cubed terms, I have five X cubed and negative X cubed. So writing those here. Moving on to my x terms, negative two x and positive eight x. And then finally, my constant that doesn't have any like terms that I'll just stick on the end there. So with my like terms grouped, we can now go ahead and combine them by adding or subtracting their coefficients, starting with our x squared terms. I have three x squared plus four x squared. Three plus four gives me seven, so that's seven x squared. Then I have five x cubed minus x cubed. Remember, that's the same thing as minus one x cubed, so five minus one is going to give me four, so plus four x cubed. Then my x term's negative two x plus eight x, that's going to be eight minus two which gives me positive six, so plus six x. And then that constant on the end there plus 10. So this is my fully simplified algebraic expression. It still is rather long. There are four terms there, but it is much less terms than what we started with. So we've simplified our expression. Let us know if you have questions, and let's keep going.

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1. Review of Real Numbers

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Beginning & Intermediate Algebra

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Simplifying Expressions

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Simplify the expression.
$3 x^{2} + 5 x^{3} - 2 x + 4 x^{2} - x^{3} + 8 x + 10$

$x^{2} - 6 x^{3} + 9 x + 10$

$x^{2} + 6 x^{3} - 9 x + 10$

$7 x^{2} + 4 x^{3} + 6 x + 10$

$- 7 x^{2} + 4 x^{3} + 6 x + 10$

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Identify and group like terms in the expression. Like terms are terms that have the same variable raised to the same power. In this expression, group the $x^3$ terms together, the $x^2$ terms together, the $x$ terms together, and the constant terms together.

Write the grouped terms explicitly: For $x^3$ terms, combine \$5x^3$ and $-x^3$; for $x^2$ terms, combine \$3x^2$ and \$4x^2$; for $x$ terms, combine $-2x$ and \$8x$; and keep the constant term $10$ as is.

Add or subtract the coefficients of the like terms within each group. For example, add the coefficients of the $x^3$ terms: $5$ and $-1$; for the $x^2$ terms: $3$ and $4$; for the $x$ terms: $-2$ and $8$.

Rewrite the expression by replacing each group of like terms with their simplified form, keeping the variable and exponent intact. This will give you a simplified polynomial expression.

Double-check your simplified expression to ensure all like terms were combined correctly and that the expression is written in standard form, usually starting with the highest power of $x$.