# Video: Dividing Third-Degree Polynomials Using Long Division

Knowing that the volume of a box is 12𝑥³ + 20𝑥² − 21𝑥 − 36, its length is 2𝑥 + 3, and its width is 3𝑥 − 4, express the height of the box algebraically.

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### Video Transcript

Knowing that the volume of a box is 12𝑥 cubed plus 20𝑥 squared minus 21𝑥 minus 36, its length is two 𝑥 plus three, and its width is three 𝑥 minus four, express the height of the box algebraically.

We assume here that this box is in the shape of a cuboid. So its volume will be equal to its length multiplied by its width multiplied by its height. We’ve been given an algebraic expression for the volume, 12𝑥 cubed plus 20𝑥 squared minus 21 𝑥 minus 36, as well as algebraic expressions for the length and width of the box, two 𝑥 plus three and three 𝑥 minus four. So we can substitute each of these expressions into the volume formula. To find the height of this box, we need to divide both sides of this equation by the expressions two 𝑥 plus three and three 𝑥 minus four, giving the quotient on the left of the screen.

We’re going to need to simplify this quotient using polynomial division. But before we do, let’s expand the brackets in the denominator. Two 𝑥 multiplied by three 𝑥 gives six 𝑥 squared. Two 𝑥 multiplied by negative four gives negative eight 𝑥. Positive three multiplied by three 𝑥 gives positive nine 𝑥. And positive three multiplied by negative four gives negative 12. We can simplify by grouping the like terms in the center of the expansion to give six 𝑥 squared plus 𝑥 minus 12. So the height of the box is equal to the quotient 12𝑥 cubed plus 20𝑥 squared minus 21𝑥 minus 36 over six 𝑥 squared plus 𝑥 minus 12. And we use algebraic division to simplify.

We set up our division with the divisor on the outside and the expression we’re dividing into on the inside. We begin by looking at the highest powers of these two expressions. And we ask ourselves, “What do we need to multiply six 𝑥 squared by to give 12 𝑥 cubed?” Well, we need to multiply six by two to give 12 and 𝑥 squared by 𝑥 to give 𝑥 cubed. So overall, we must multiply by two 𝑥.

We now multiply the full expression six 𝑥 squared plus 𝑥 minus 12 by two 𝑥. Six 𝑥 squared multiplied by two 𝑥 gives 12𝑥 cubed, as already discussed. 𝑥 multiplied by two 𝑥 gives two 𝑥 squared. And negative 12 multiplied by two 𝑥 gives negative 24𝑥. Next, we subtract this expression from the original expression. 12𝑥 cubed minus 12𝑥 cubed gives zero. 20𝑥 squared minus two 𝑥 squared gives 18𝑥 squared. And negative 21𝑥 minus negative 24𝑥 becomes negative 21𝑥 plus 24𝑥, which is positive three 𝑥. We also have negative 36 minus zero. So, we still have negative 36.

We still have a remainder at this point. So, we keep going. And again, we look at the highest powers, asking ourselves, “What do we multiply six 𝑥 squared by to give 18𝑥 squared?” We must multiply by three. Six multiplied by three is 18. So six 𝑥 squared multiplied by three is 18𝑥 squared. Now we multiply the full expression by three. Six 𝑥 squared multiplied by three gives 18𝑥 squared. 𝑥 multiplied by three gives plus three 𝑥. And negative 12 multiplied by three gives negative 36.

As before, we subtract in columns. 18𝑥 squared minus 18𝑥 squared gives zero. Three 𝑥 minus three 𝑥 gives zero. And negative 36 minus negative 36 — that’s negative 36 plus 36 — is also zero. So, we have no remainder. And we’ve therefore reached the end of our algebraic division. In total, we multiplied six 𝑥 squared plus 𝑥 minus 12 by two 𝑥 and then by positive three. So overall, we’ve multiplied by two 𝑥 plus three. This tells us that the answer to the algebraic division 12𝑥 cubed plus 20𝑥 squared minus 21𝑥 minus 36 divided by six 𝑥 squared plus 𝑥 minus 12 is two 𝑥 plus three. And so this is the height of this box expressed algebraically. The height is two 𝑥 plus three.