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How Do You Solve a Word Problem by Subtracting and Multiplying Polynomials?

Leroy used to only grow corn in his garden, but recently he decided he wanted to also plant some beets. He made a blueprint for how he was going to organize his garden, which is given in the diagram box on the right. Write a polynomial in simplest form for the area that he will have left to grow corn.

Summary

  1. The area for corn is the total area of the garden minus the area for beets
  2. The total area of the garden is its width, 3a, times its length, 2a2+4a-5
  3. The area of the beet section is its width, 4, times its length, a2-3a+2
  4. Distribute 3a through the first polynomial
  5. Distribute 4 through the second polynomial
  6. Distribute the negative through the second polynomial

Notes

    1. We don't really have a good equation for a funny-shaped area like that
    1. But we can use the areas of the entire garden and the section for beets to figure it out
    2. Since those sections are just rectangles it will be much easier
    1. We can use the areas of the entire garden and the beet section to find the area of the corn section
    1. The length of the entire garden is 2a2+4a-5
    2. The width of the entire garden is 3a
    3. The length of the section for beets is a2-3a+2
    4. The width of the section for beets is 4
    1. The area of the entire garden is its length times its width
    2. So it will be 2a2+4a-5 times 3a
    3. The area of the beet section is its length times its width
    4. So it will be a2-3a+2 times 4
    5. Subtracting these two areas will give us the area left for corn
    1. It's just the area of the entire garden minus the area for beets
    1. 2a2+4a-5 is the length of the entire garden
    2. 3a is the width of the entire garden
    1. a2-3a+2 is the length of the beet section
    2. 4 is the width of the beet section
    1. Now that we have values for our areas, we can subtract them to find the area for the corn
    1. The first step in simplifying our equation is multiplying out the areas
    1. Distribute the 3a through the first polynomial
    1. Distribute the 4 through the second polynomial
    1. Now that we've finished multiplying, we need to add our polynomials together
    1. The first polynomial is 6a3+12a2-15a
    2. Since there is no negative out front, we can just drop the parentheses and rewrite it
    1. The second polynomial is 4a2-12a+8
    2. Since there is a negative in front of the parentheses, we need to distribute the negative before we can drop the parentheses
    1. So 4a2 becomes -4a2
    2. -12a becomes +12a
    3. And 8 becomes -8
    1. We need to combine like terms to put our polynomial in simplest form
    2. 6a3 is the only term with an a3, so just rewrite it
    3. 12a2 and 4a2 both have an a2, so subtract them to get 8a2
    4. -15a and 12a both have an 'a', so add them to get -3a
    5. -8 is the only term without a variable, so just rewrite it
    1. Our polynomial is now in simplest form
    2. The area left for corn in Leroy's garden is the polynomial 6a3+8a2-3a-8