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A rectangle has an area of 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10 square centimeters and a width of 𝑦 plus two centimeters.
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Find its length in terms of 𝑦 and its perimeter when 𝑦 equals four.
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Well the area of a rectangle is calculated by multiplying the length by the width.
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If we let the length of the rectangle equal 𝑥 centimeters, then 𝑥 multiplied by 𝑦 plus two must be equal to 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10.
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Dividing both sides by 𝑦 plus two allows us to calculate the length by dividing 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10 by 𝑦 plus two.
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We are going to now use long division to work out the length of the rectangle.
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In order to divide 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10, a cubic, by 𝑦 plus two, a linear expression, we need to divide, multiply, and subtract.
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Firstly, we need to divide 𝑦 cubed by 𝑦.
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This gives us 𝑦 squared.
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Secondly, we need to multiply this 𝑦 squared by 𝑦 plus two to work out our remainder.
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𝑦 squared multiplied by 𝑦 is 𝑦 cubed, and 𝑦 squared multiplied by two is two 𝑦 squared.
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Our next step is to subtract those two lines.
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Well 𝑦 cubed minus 𝑦 cubed is zero.
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And in this case, two 𝑦 squared minus two 𝑦 squared is also zero.
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Therefore, there is no remainder.
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This means that we do not need to divide the 𝑦 squared term by 𝑦.
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So we can drop down the five 𝑦 plus 10.
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We now repeat the process by dividing five 𝑦 by 𝑦.
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Well, as five 𝑦 divided by 𝑦 is equal to five, we can put five on the answer line.
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Multiplying this five by 𝑦 plus two gives us five 𝑦 plus 10 as five multiplied by 𝑦 is five 𝑦 and five multiplied by two is 10.
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When we subtract these two terms again, we can see that once again there is no remainder as five 𝑦 minus five 𝑦 is zero and 10 minus 10 is also zero.
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Therefore, we can say that 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10 divided by 𝑦 plus two is equal to 𝑦 squared plus five.
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Therefore, the length of the rectangle is 𝑦 squared plus five.
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At the end of this video, I will show you an alternative method for calculating the length in terms of 𝑦.
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However, we will now focus on the second part of the question: finding the perimeter.
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As we now know both of width and the length of the rectangle, we can substitute in the value 𝑦 equals four to calculate its perimeter.
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The width of the rectangle is four plus two, which is six centimeters.
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And the length is four squared plus five, which is 21 centimeters, as four squared is equal to 16 and 16 plus five equals 21.
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To calculate the perimeter of any shape, we need to work out the distance around the outside of the shape.
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In this case, we need to add 21, six, 21, and six.
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Adding these four numbers gives us 54 centimeters.
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Therefore, the perimeter of the rectangle is 54 centimeters.
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Therefore, if a rectangle has an area of 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10 and a width of 𝑦 plus two, its length is 𝑦 squared plus five.
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And the perimeter when 𝑦 equals four is 54 centimeters.
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We will now look at an alternative method for finding an expression for the length in terms of 𝑦.
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As the area of a rectangle is calculated by multiplying the width by the length, we know in this case that 𝑦 plus two multiplied by something is equal to 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10.
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As 𝑦 plus two is a linear equation and 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10 is a cubic equation, the missing term, in this case the length, must be a quadratic expression as a linear expression multiplied by a quadratic expression gives us a cubic expression.
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Considering the general quadratic expression 𝑎𝑦 squared plus 𝑏𝑦 plus 𝑐, we can see that 𝑦 plus two multiplied by this quadratic expression gives us 𝑦 cubed plus two 𝑦 squared plus five 𝑦 plus 10.
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Expanding on multiplying out the brackets or parentheses gives us 𝑎𝑦 cubed plus 𝑏𝑦 squared plus 𝑐𝑦 when we multiply the quadratic by 𝑦 and two 𝑎𝑦 squared plus two 𝑏𝑦 plus two 𝑐 when we multiply the quadratic by two.
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We know that this must be equal to the cubic expression for our area.
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Comparing coefficients gives us four equations.
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For the 𝑦 cubed terms, 𝑎 equals one.
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For the 𝑦 squared terms, 𝑏 plus two 𝑎 equals two.
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For the 𝑦 terms, 𝑐 plus two 𝑏 equals five.
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And for the free terms, two 𝑐 equals 10.
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Well the first equation tells us that 𝑎 is equal to one.
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And dividing the fourth equation by two tells us that 𝑐 is equal to five.
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We can now substitute either these two values into the second or third equation to work out the value of 𝑏.
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Well as five plus two 𝑏 is equal to five, 𝑏 must be equal to zero.
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Substituting these back into our quadratic expression gives us one 𝑦 squared plus nought 𝑦 plus five.
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This can be simplified to give us the expression for the length of 𝑦 squared plus five.
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These are just a couple of different ways of finding the length in terms of 𝑦.