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Knowing that the length of a rectangle is three π₯ minus four and its area is six π₯ to the fourth minus eight π₯ cubed plus nine π₯ squared minus nine π₯ minus four, express the width of the rectangle as a polynomial in standard form.
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Here we have a rectangle, its length is equal to three π₯ minus four.
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We are solving for the width.
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And then our area is equal to this polynomial.
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Length times width is equal to the area.
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So if we are solving for the width, which will be π, we need to divide both sides by πΏ.
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This means, in order to find the width, we need to take the area and divide by the length.
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So here we can solve by dividing, and we can use something called long division.
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So we begin by looking at the first terms.
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What do we multiply to three π₯ so it looks like six π₯ to the fourth?
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We will need to multiply it by two π₯ cubed, so we put that above the cubed term.
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And now we distribute.
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And now we subtract.
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So all of these cancel.
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So this means we need to bring down the nine π₯ squared.
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And now we start over.
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How do we get three π₯ to look like nine π₯ squared?
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What do we multiply by?
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And that would be three π₯, we put above the π₯ term.
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We can always add in a zero for something that isnβt being used.
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So for here, thatβll be zero π₯ squared.
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We do not need to include it in our final answer.
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So now we distribute the three π₯.
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Now after weβve distributed, we need another term, so we bring down the negative nine π₯.
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And now we subtract.
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So we have negative nine π₯ minus negative 12π₯, so itβs really plus 12π₯.
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So we get three π₯, and we start the process over again.
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How do we get three π₯ to look like three π₯?
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What do we multiply by?
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And that would be one.
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So we distribute, bring down the minus four, and subtract.
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And we get zero.
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So this means our remainder is zero.
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So this means our width is two π₯ cubed plus three π₯ plus one.