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Question Video: Simplifying Polynomials by Division to Express the Length of a Rectangle Mathematics • 10th Grade

What is the length of a rectangle whose area is (6π‘₯⁡𝑦 + 27π‘₯⁴𝑦) cmΒ² and whose width is π‘₯𝑦 cm?

03:50

Video Transcript

What is the length of a rectangle whose area is six π‘₯ raised to the fifth power 𝑦 plus 27π‘₯ raised to the fourth power 𝑦 square centimeters and whose width is π‘₯𝑦 centimeters?

In this question, we are asked to determine the length of a rectangle from an expression for its area in square centimeters in terms of unknowns π‘₯ and 𝑦 and an expression for its width in the same unknowns in centimeters. To answer this question, we can start by sketching the given information onto a rectangle. We can call the length of the rectangle 𝐿 centimeters, and we can add the expressions for the area and width to obtain the following sketch.

We can then recall that the area of a rectangle is equal to its length multiplied by its width. We can substitute the given expressions for the area and width of the rectangle to form an equation. We have six π‘₯ raised to the fifth power times 𝑦 plus 27π‘₯ raised to the fourth power 𝑦 is equal to 𝐿 times π‘₯𝑦. Since we want to find an expression for the length of the rectangle, we want to isolate 𝐿 on one side of the equation. To do this, we need to divide the equation through by π‘₯𝑦.

Before we do this, it is worth noting that both π‘₯ and 𝑦 are nonzero, since if either value was zero, then the width of the rectangle would be zero centimeters. Therefore, we can divide both sides of the equation through by π‘₯𝑦 to obtain six π‘₯ raised to the fifth power 𝑦 plus 27π‘₯ raised to the fourth power 𝑦 all over π‘₯𝑦 is equal to 𝐿.

We can now note that on the left-hand side of the equation, we are dividing a polynomial by a monomial. To simplify this division, we want to divide every term in the numerator by the denominator separately. Splitting the division over each term gives us the following equation. It is worth noting that we can simplify the left-hand side of the equation by using the quotient rule for exponents. However, we can start by canceling the shared factor of 𝑦 in the numerator and denominator of each term, since we know that 𝑦 is nonzero.

We can follow a similar process to simplify the division by π‘₯. Or we can apply the quotient rule for exponents, which tells us π‘₯ raised to the power of π‘š over π‘₯ raised to the power of 𝑛 is equal to π‘₯ raised to the power of π‘š minus 𝑛, provided that π‘₯ is nonzero. We can rewrite the π‘₯ in the denominator as π‘₯ raised to the first power so that we can apply this rule to each term. We obtain six π‘₯ raised to the power of five minus one plus 27π‘₯ raised to the power of four minus one is equal to 𝐿. Finally, we can evaluate the expressions in the exponents and add the units of centimeters onto our length to conclude that the length of the rectangle is six π‘₯ raised to the fourth power plus 27π‘₯ cubed centimeters.

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