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[MUSIC] Let's learn how to divide polynomials.

Â [SOUND] For example, let's divide x^2 + 9x + 19 by x + 4.

Â Now we begin dividing polynomials in a similar way as we do when we divide

Â numbers. But we start by looking at the leading

Â terms. And ask ourselves, x times what is equal

Â to x^2. And this would be x wouldn't it? Because

Â x^2 / x = x. So that's the first term in our quotient

Â here, x.

Â And now we need to multiply x by the entire divisor x + 4 which gives us x^2 +

Â 4x. And then just like when we divide

Â numbers, we now subtract x^2 - x^2 is 0. 9x - 4x is 5x and we still have this + 19

Â here. Now are we done? We're not because the

Â degree of this is not less than the degree of this.

Â So we continue until the degree is smaller than the degree of the divisor.

Â Again, we look at the leading terms here, x and 5x.

Â And ask ourselves, x times what is equal to 5x. And this would be 5, wouldn't it?

Â Because 5x / x = 5. So that's the next term in our quotient.

Â So we have + 5, and now we multiply 5 by the entire divisor x + 4 which gives us

Â 5x + 20. Again, we'll subtract 5x - 5x = 0, and 19

Â - 20 = -1. And now the degree of -1 is 0, which is

Â smaller than the degree of the divisor, so we are done.

Â Well how can we represent our answer here?

Â By the division algorithm, we have that the dividend x^2 + 9x + 19 divided the

Â divisor x + 4 = the quotient x + 5 + the remainder which is -1 / the divisor x + 4

Â or we can multiply both sides of the equation by the divisor x + 4 which gives

Â us that x^2 + 9x + 19 = x + 5 * x + 4 and then - 1.

Â So these are two nice ways of representing our answer.

Â And in this last form here, we can actually check that we've done this

Â division correctly by foiling this out. And then subtracting 1.

Â So let's do that, when we foil out the right side we get x^2 + 4x + 5x + 20 and

Â then we still have the -1 which is equal to x^2 + 9 x + 19 which sure enough is

Â our dividend. All right, let's look at another example.

Â [SOUND] Let's divide. Well the first thing to notice here is

Â that our dividend Is not written in standard form.

Â Standard form would be 4x^4 - 11x^2 + 15x + 7.

Â And the other thing to notice, is also there's no x^3 term.

Â So let's add a placeholder term with a coefficient of 0.

Â That is, let's write our dividend 4x^4 + 0x^3 - 11x^2 + 15x + 7.

Â Alright, so we're ready to divide now. Our divisor is 2x^2 + 3x - 2.

Â Our divided is 4x^4 + 0x^3 - 11x^2 + 15x + 7.

Â Again we start by looking at the leading terms.

Â Which is why we wanted to rewrite this dividend in the first place.

Â So we're going to ask ourselves, 2x^2 times what is equal to 4x^4.

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And, then we still have this +15x + 7. Now we didn't need to write this place

Â holder here, this 0x^3 but when we subtract the 6x^3, it comes in handy

Â because then we know we are subtracting 6x^3 from 0x^3.

Â All right, now we have to ask ourselves how many times this goes into this?

Â That is, 2x^2 * what = -6x^3. And that would be -3x, wouldn't it?

Â Because -6x^3 / 2x^2 = -3x which is the next term in our quotient.

Â So, we have a -3x and then -3x times our entire divisor, give us -6x3 - 9x^2 + 6x.

Â Again, we subtract -6x^3 - -6x^3 = 0, and then -7x^2 - -9x^2 is +2x^2.

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Then 15x - 6c is +9x and then we still have the +7.

Â All right, 2x^2 goes into 2x^2 one time. So, that would be the last term in our

Â quotient here. So, 1 times our divisor is 2x^2 + 3x - 2.

Â Again, we subtract 2x^2 - 2x^2 is 0, 9x - 3x = 6x, 7 - -2 is +9.

Â And now the degree of this is smaller than the degree of our divisor up here,

Â so we are finished. So, by the division algorithm this dividend divided by the

Â divisor is equal to the quotient, just 2x^2 - 3x + 1 + the remainder 6x+9

Â divided by the divisor, this 2x^2 + 3x - 2.

Â