Perform the indicated operation. ( − 2 x 4 + 10 x 3 + 6 x − 3 ) − ( x 4 − 7 x 2 + 8 x + 5 ) \left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right) ( − 2 x 4 + 10 x 3 + 6 x − 3 ) − ( x 4 − 7 x 2 + 8 x + 5 )

Hey, everyone. Welcome back. In this problem, we have a really long polynomial that's subtracting another very long polynomial. So let's go ahead and get started here. Remember that before you can start combining like terms, you have to look to see if there's subtraction signs that are outside of parentheses. Because remember, this sign over here gets distributed to everything that's in front of it in the parenthesis. Be very careful. Don't just start, you know, combining your like terms. So let's go ahead and do that first. There's nothing that's gonna happen to this expression over here, so I can actually just drop the parenthesis. 2x4th10x3rd, and this is 6x-3. What happens over here? The negative into the x to the 4th power becomes negative x to the 4th. The negative into the negative 7 x squared, always be careful because I have negative distributing into something that's already negative, becomes positive 7 x squared. Now what happens here is I have a negative 8 x and then I have a negative 5 over here. So all these things basically just flip their signs. Okay? Now that what happens is now that we've distributed the minus sign, now we can just drop the parentheses, and now we can get started on combining our like terms. So let's go ahead and do that. So do I have any x to the 4th powers? I see x to the 4th then x to the 4th over here. What about x to the thirds? I have an x to the 3rd power here, and then I don't have anything else. What about x squared? I've got one x squared over here. What about x's? Let's see. I've got one here and then I've got one here. So I've got 6x and negative 8x, And then finally about constants too, so I've got negative 3 and negative 5. Alright? I still don't have that many highlighter colors. Anyway, so let's go ahead and and subtract down. We've got negative 2 x and then negative x to the 4th. So I have to combine those. Negative 2 and negative one become negative 3 x to the 4th. So I'm done with that. Now what about my x square x cubed terms? My x cubed terms become I have 10x cubed over here, and then I have a that's it. That's so that's the only thing that happens. So, I just do 10x cubed, and that's plus. So it doesn't combine with anything. What about 7 x squared? That's also by itself. So I just drop that straight down. They have that that becomes 7 x squared. And then finally, I think I've got let's see. I've got some 6 x's over here and then I've got an a negative 8 x. So 6 x and negative 8 x will actually combine down. Let's put this over here. Those those will combine down to form, negative 2x. And then finally, I've got negative 3 and negative 5. Those will combine down to form negative 8. Okay. So now I just got to I just gotta double check this real quickly. Do I see if you know, can I combine any other like terms here? And I don't think I can because I've got 3x4th. Basically, every power just pops up only once, so I can't combine any more like terms. Alright? So that is how to subtract these 2 polynomials. Just always remember to distribute that negative sign. Alright? That's it for this one. Thanks for watching.

Perform the indicated operation. ( − 2 x 4 + 10 x 3 + 6 x − 3 ) − ( x 4 − 7 x 2 + 8 x + 5 ) \left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right) ( − 2 x 4 + 10 x 3 + 6 x − 3 ) − ( x 4 − 7 x 2 + 8 x + 5 )

− 3 x 4 + 10 x 3 + 7 x 2 − 2 x − 8 -3x^4+10x^3+7x^2-2x-8 − 3 x 4 + 10 x 3 + 7 x 2 − 2 x − 8

− 3 x 4 + 17 x 3 − 2 x − 8 -3x^4+17x^3-2x-8 − 3 x 4 + 17 x 3 − 2 x − 8

− 3 x 4 + 17 x 2 − 2 x − 8 -3x^4+17x^2-2x-8 − 3 x 4 + 17 x 2 − 2 x − 8

− x 4 + 10 x 3 − 7 x 2 + 14 x + 2 -x^4+10x^3-7x^2+14x+2 − x 4 + 10 x 3 − 7 x 2 + 14 x + 2

0. Fundamental Concepts of Algebra

Polynomials Intro

Precalculus

0. Fundamental Concepts of Algebra

Polynomials Intro

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Multiple Choice

Perform the indicated operation.
$\left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right)$

$-3x^4+10x^3+7x^2-2x-8$

$-3x^4+17x^3-2x-8$

$-3x^4+17x^2-2x-8$

$-x^4+10x^3-7x^2+14x+2$

Verified step by step guidance

First, distribute the negative sign across the second polynomial, changing the signs of each term: $-(x^4 - 7x^2 + 8x + 5)$ becomes $-x^4 + 7x^2 - 8x - 5$.

Rewrite the expression by combining the two polynomials: $(-2x^4 + 10x^3 + 6x - 3) + (-x^4 + 7x^2 - 8x - 5)$.

Combine like terms: Start with the highest degree term, which is $x^4$. Combine $-2x^4$ and $-x^4$ to get $-3x^4$.

Next, combine the $x^3$ terms. There is only one $x^3$ term, which is $10x^3$, so it remains $10x^3$.

Combine the $x^2$, $x$, and constant terms: $7x^2$ (since there is no $x^2$ term in the first polynomial), $6x - 8x = -2x$, and $-3 - 5 = -8$.

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Struggling with Precalculus?

Perform the indicated operation.(−2x4+10x3+6x−3)−(x4−7x2+8x+5)\left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right)(−2x4+10x3+6x−3)−(x4−7x2+8x+5)

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Polynomials Intro practice set

Perform the indicated operation.
$\left(-2x^4+10x^3+6x-3\right)-\left(x^4-7x^2+8x+5\right)$