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How Do You Use a Polynomial Equation to Solve a Word Problem?

Peter just moved to a new apartment. His new bedroom has the same perimeter as his old bedroom. The diagram below shows a blueprint for the dimensions of his old bedroom and his new bedroom in inches. Find the perimeter of his bedroom.

Summary

  1. Perimeter is the sum of all the sides
  2. The two perimeters are equal
  3. Factor out the GCF, 2xy
  4. Group together terms with common factors
  5. Split up the factors and set each one equal to 0
  6. Dimensions can't be 0, so x=7 and y=3 are our solutions

Notes

    1. Remember, perimeter is the distance around the edge of a shape
    2. Since the perimeters are the same, we will be able to set them equal to each other
    1. This was given on the blueprint
    2. The length of the old bedroom was 2x2y2
    3. The width of the old bedroom was 42xy
    4. The length of the new bedroom was 6x2y
    5. The width of the new bedroom was 14xy2
    1. Remember, perimeter is the distance around the edge of a shape
    2. So to get the perimeter of each bedroom we just need to add up the sides
    3. The problem tells us the perimeters are equal, so we can set them equal to each other
    1. The length of the old bedroom was 2x2y2
    2. The width of the old bedroom was 42xy
    3. We could also add 42xy+42xy+2x2y2+2x2y2, but it's easier to add the two sides and multiply by 2
    1. The length of the new bedroom was 6x2y
    2. The width of the new bedroom was 14xy2
    3. We could also add 6x2y+6x2y+14xy2+14xy2, but it's easier to add the two sides and multiply by 2
    1. Division is the opposite of multiplication, so dividing by 2 on both sides undoes the multiplication
    1. Our equation will be easier to solve if we have all the terms on one side
    1. Subtraction is the opposite of addition
    2. So subtracting these terms will cancel them out on the left and move them to the right
    1. Now our equation will be easier to solve
    1. We need to see if we can factor out a greatest common factor from each of our terms
    1. Our coefficients are 6, 14, -42, and -2
    2. These have a common factor of 2
    1. Each of our terms has at least one x
    2. So x is a common factor
    1. Each of our terms has at least one y
    2. So y is a common factor
    1. Multiply together the common factors 2, x, and y
    2. This gives us a greatest common factor of 2xy
    1. 6/2=3, and we can cancel an x and a y from 6x2y, leaving 3x
    2. 14/2=7, and we can cancel an x and a y from 14xy2, leaving 7y
    3. 42/2=21, and we can cancel an x and a y from 42xy, leaving 21
    4. 2/2=1, and we can cancel an x and a y from 2x2y2, leaving xy
    1. We've already factored out the greatest common factor, but we can still factor the polynomial further
    1. We can group together the terms that have common factors
    1. 3x and xy have a common factor of x
    2. 7y and -21 have a common factor of 7
    1. In the first set the common factor is x
    2. In the second set the common factor is 7
    1. x is a common factor
    2. The x's cancel in the first term, leaving just 3
    3. The x's cancel in the second term, leaving just y
    4. So we have an x outside the parentheses and 3-y inside
    1. We want to factor a -7 so that we get the same thing inside the parentheses as we do in the first term
    2. When we factor out a negative, it will switch the signs of the terms we factored
    3. Factoring a -7 from 7y cancels the 7 and flips the sign, leaving us with -y
    4. Dividing two negatives gives a positive, so dividing -21/-7 gives us 3
    1. Now each of our terms has a 3-y, so we can factor the whole thing out
    1. Bring the 3-y out to the front and leave the x-7 in parentheses
    2. Now our polynomial is completely factored!
    1. If only that meant we were done!
    1. Now we can use the zero-product property to find our solutions
    1. We are multiplying three things together to equal 0
    2. So the zero product property says that one of those three things must equal 0
    3. We can use this fact to find our solutions
    1. Our factors are 2xy, 3-y, and x-7
    2. Set each of these factors equal to 0
    1. We can apply the zero-product property again here
    1. 2xy has 2 times x times y equal to 0
    2. So one of the factors must equal 0
    1. We already know that 2 doesn't equal 0, so we can ignore it
    2. But either x or y could equal 0 to give us an answer of 0
    3. So x=0 and y=0 are two possible solutions to our equation
    1. To solve our equation for y, undo the subtraction by adding y to both sides
    2. So y=3 is another possible solution that would make our factor 0
    1. To solve our equation for x, undo the subtraction by adding 7 to both sides
    2. So x=7 is another possible solution that would make our factor 0
    1. But we're still not done yet!
    1. Our problem wasn't asking us to just solve the equation
    2. We still need to find the perimeter
    1. Take the width of his old bedroom, 42xy, for example
    2. If we plug in 0 for x or y, we'll get a width of 0
    3. It's not possible for Peter's bedroom to have a width of 0!
    4. So we can rule out x=0 and y=0 as possible solutions
    1. This leaves us with just x=7 and y=3 as our solutions
    1. We're a little closer but we're still not done yet - we still need to find the perimeter
    1. We need to plug in 7 for x and 3 for y our original perimeter equation
    1. The perimeter of the old bedroom was 2(42xy+2x2y2)
    2. The perimeter of the new bedroom was 2(6x2y+14xy2)
    3. Since these two perimeters are equal, we should get the same answer if we plug x=7 and y=3 into either
    1. So let's choose the old bedroom
    2. We'll plug x=7 and y=3 into this expression
    1. 7 replaces x and 3 replaces y
    2. Then simplify, using order of operations:
    3. 7•3=21
    4. 72=49
    5. 32=9
    6. 42•21=882
    7. 2•49•9=882
    1. Remember to do the addition in the parentheses first!
    2. 882+882=1764
    3. Then multiply by 2
    4. 1764•2=3528
    1. Remember, our perimeter was in inches