Divide using long division. Write the result as dividend = (divisor)(quotient) + remainder. (x^3

Divide using long division. Write the result as dividend =...

Key Concepts

Polynomial Long Division

A systematic method used to divide a polynomial by another polynomial of lower degree. It involves dividing the highest degree terms, multiplying the divisor by the appropriate term to subtract from the dividend, and then repeating these steps with the remainder until reaching a polynomial with a smaller degree than the divisor.

Division Algorithm for Polynomials

A fundamental concept that states for any two polynomials, where the divisor is not zero, there exist unique quotient and remainder polynomials such that the dividend equals the divisor multiplied by the quotient plus the remainder, with the degree of the remainder being less than the degree of the divisor.

Dividend, Divisor, Quotient, and Remainder in Polynomial Division

These are the key components in any division process involving polynomials. The dividend is the polynomial being divided, the divisor is the polynomial by which we divide, the quotient is the result of the division, and the remainder is what is left over. Together, they satisfy the relationship established by the division algorithm for polynomials.

Transcript

00:01 Okay, we're going to tackle another division problem with polynomials here.

00:05 We're going to solve this problem using long division, and we're going to provide the answer in the format that we have on the top right corner, also highlighted in yellow.

00:14 So let's get started.

00:17 First, we need to find the multiplier.

00:19 In this problem, our multiplier will be the result of the division of the first term of the dividend, which is x cubed, divided by the first term of the divide.

00:31 Which is x.

00:33 Of course we get x squared.

00:36 This will become our multiplier.

00:40 So we can place it here where the quotient goes.

00:43 Now we can move on to the second step, which is multiply the multiplier by the divisor.

00:48 X squared times x is x cubed.

00:51 X squared times 2 is 2x squared.

00:55 Now before we can actually subtract, we have to move on to the third step, which is distributing the negative one.

01:05 Now we've got to distribute the negative 1 to both terms, which is x cubed and 2x squared.

01:10 So negative 1 times x cubed is negative x cubed.

01:14 2x squared times negative 1 is negative 2x squared.

01:19 And now we can actually start subtracting.

01:22 I mean, we can get rid of this now.

01:24 We don't need that anymore.

01:26 Now x cubed minus x cubed, 0.

01:29 Negative 5x squared minus 2x squared is negative 7x squared.

01:34 Now we can bring down the second part.

01:38 And we begin the process all over again.

01:41 First we need to find the multiplier.

01:43 Now we treat this as the new dividend.

01:50 Every time we finish subtracting, whatever we get we can treat as our new dividend.

01:56 So we'll take the first term, which is negative 7x squared, and divide it by the first term of our divisor, x, and we get negative 7x because x squared divided by x, well one of those goes away.

02:11 Negative 7x is now here in our quotient now we can move on to the second step negative 7x times x is negative 7x squared negative 7x times 2 negative 14 x now we can move on to the third step which is to subtract but don't forget to distribute the negative 1 we get 7x squared plus 14x once we distribute.

02:44 And after we distribute, we can now go ahead and start subtracting...

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