Multiply the polynomials using special product formulas.(2x+4)3\left(2x+4\right)^3(2x+4)3

8 x 3 + 48 x 2 + 96 x + 64 8x^3+48x^2+96x+64 8 x 3 + 48 x 2 + 96 x + 64

Multiply the polynomials using special product formulas. ( 2 x + 4 ) 3 \left(2x+4\right)^3 ( 2 x + 4 ) 3

Hey, everyone. So in this practice problem, I've got the expression 2x+4, and that whole expression is cubed. And I wanna do this wanna see if I have multiplied this by using special product formulas because it's gonna be way easier than if I have to foil and multiply everything out manually. So I've got my formulas over here for special products. Just have to figure out which one of these things it sort of applies to. Is it the a plus b, a minus b? Is it a plus b squared or is it a plus b cubed? Well, should be, you know, pretty clear here that it's not gonna be a difference of squares. I don't see a plus b a minus b. So it's really just gonna be one of these two's expressions where you have a parenthesis and it's raised to a power. And if you look at this, what happens is this is an expression that's squared, so it's not gonna be this one. And in fact, it's actually gonna be this one over here. So it's a plus b cubed. Alright. So this really is like a plus b, and this is gonna be cubed. And I'm actually gonna do this over here because it it'll be a little bit neater. I have a+b, this is cubed. So what is the answer? Well, remember how this is how this is gonna look in our polynomials is if you have a plus sign here, then this will be plus. In fact, actually, everything will be plus, so that at least makes our sign a little bit easier. So this is gonna become a3 plus and remember how the a's work, they go decreasing, the coefficients are 1331. So I have 3 and then one over here, and then what happens is I've had a squared b a b squared, and then I have just b squared. Alright. So now what I have to do here is I'm just gonna have to go calculate what a, b, and all the other powers are. Alright? So here's what happens. I've got a and b. So my a here is 2x, and my b in this case is equal to, just 4. So what happens here is I've got a squared. A squared is 2x squared, so that's gonna be 4x squared. And then a cubed is gonna be if I take 2x and multiply by itself that's 4x squared and my 2x2x4x squared that just becomes 8x cubed. Right? So it's that whole entire expression. You multiply the 2 by itself 3 times, the x by itself 3 times. And then what happens over here is that I've got b squared is equal to 16, and then b cubed is just 4 times 16, and that's just 64. So I basically got all of my, you know, a's and all the powers of a and b and all that stuff. Now I'm just gonna go ahead and pop them into this equation over here, and I might have to simplify a few things. So I've got a cubed is my first term, that's the a x squared. So that's 8 x squared. Then I've got 3 a squared times b. So this is 3 times a squared, which is 4x squared times b, which is just 4. Then my 3rd term has 3ab squared. So a is 2x, and b squared is 16. So that's 16. And then I finally got b oh, I'm sorry. This is actually supposed to be b to the third power. If you caught that, great job. It's supposed to be b cubed. And b cubed over here just becomes 64. So that is plus 64. Alright. So let's bring this down here and clean this up a little bit. My final answer is gonna be this is gonna be 8x3 plus I've got 3 times 4, so that becomes 12, and 12 times 4x squared becomes 48x squared. Now over here, what I've got here is I've got 3 times 16. 3 times 16 is also 48, but now you're gonna do 48 times 2x, and that becomes 96x. And then finally, the 64 doesn't change. That should become 64. And that's our answer. So I've got that expression over here. Can't combine any more like terms, and that's how to take a polynomial like 2x+4 and raise it to the 3rd power using special formulas. Thanks for watching.

Multiply the polynomials using special product formulas. ( 2 x + 4 ) 3 \left(2x+4\right)^3 ( 2 x + 4 ) 3

8 x 3 + 48 x 2 + 96 x + 64 8x^3+48x^2+96x+64 8 x 3 + 48 x 2 + 96 x + 64

8 x 3 + 16 x 2 + 32 x + 64 8x^3+16x^2+32x+64 8 x 3 + 16 x 2 + 32 x + 64

8 x 3 + 24 x 2 + 24 x + 64 8x^3+24x^2+24x+64 8 x 3 + 24 x 2 + 24 x + 64

x 3 + 24 x 2 + 96 x + 64 x^3+24x^2+96x+64 x 3 + 24 x 2 + 96 x + 64

0. Fundamental Concepts of Algebra

Multiplying Polynomials

Precalculus

0. Fundamental Concepts of Algebra

Multiplying Polynomials

Struggling with Precalculus?

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Multiple Choice

Multiply the polynomials using special product formulas.
$$\left$(2x+4$\right$)^3$

$8x^3+16x^2+32x+64$

$8x^3+24x^2+24x+64$

$8x^3+48x^2+96x+64$

$x^3+24x^2+96x+64$

Verified step by step guidance

Recognize that the expression $(2x + 4)^3$ is a perfect cube and can be expanded using the binomial theorem or special product formulas.

Recall the binomial expansion formula for a cube: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. Here, $a = 2x$ and $b = 4$.

Substitute $a = 2x$ and $b = 4$ into the formula: $(2x)^3 + 3(2x)^2(4) + 3(2x)(4)^2 + 4^3$.

Calculate each term separately: $(2x)^3 = 8x^3$, $3(2x)^2(4) = 3 \times 4x^2 \times 4 = 48x^2$, $3(2x)(4)^2 = 3 \times 2x \times 16 = 96x$, and $4^3 = 64$.

Combine all the terms to form the expanded polynomial: $8x^3 + 48x^2 + 96x + 64$.

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Multiply the polynomials using special product formulas.(2x+4)3\(\left\)(2x+4\(\right\))^3(2x+4)3

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Multiply the polynomials using special product formulas.
$\(\left\)(2x+4\(\right\))^3$