Some products of polynomials appear frequently in real-life problems. Expressions like (a + b)2, (a – b)2, and (a + b)(a – b) occur so often that it is helpful to develop patterns for their products.
To develop these patterns, we will model a simpler expression, (x + 1)2, geometrically.
The area of a square with a side length of x + 1 is (x
+ 1)(x + 1) or (x + 1)2. The total area can also
be found by adding the areas of the inner regions together. The right side of
the equation below represents the area of the square as the sum of the areas of
the four small squares.
You
can use a similar model to find (x
– 1)2. x – 1 = x + (–1).
The square of a sum and the square of a difference can be
found by using the following rules.
Find each product.
1.
(x + 4)2
Alternative Solutions:
2.
(4m + n)2
Alternative Solutions:
3. (w – 5)2
Alternative Solutions:
4.
(3p – 2q)2
Alternative Solutions:
Biologists use a method that is similar to squaring a sum to
find the characteristics of offspring based on genetic information.
Biology
Link
5. In
a certain population, a parent has a 10% chance of passing the gene for brown
eyes to its offspring. If an offspring receives one eye-color gene from its
mother and one from its father, what is the probability that an offspring will
receive at least one gene for brown eyes?
Alternative Solutions:
There is a 10% chance of passing the gene for brown eyes.
Therefore, there is a 90% chance of not passing the gene.
Use the model at the right to show all possible combinations. In the model, B represents the gene for brown eyes and b represents the gene for not brown eyes. Note that the percents are written as decimals.
Three
of the four small squares in the model contain a B. Add their probabilities.
0.01 + 0.09 + 0.09 = 0.19
So, the
probability of an offspring receiving at least one gene for brown eyes is 0.19
or 19%.
You can use the FOIL method to find product of the sum and
difference of the same two terms. Consider (x + 3)(x – 3).
Note that the product is the difference of the squares of the
terms. The product of a sum and a difference can be found by using the following
rule.
Example
Find each product.
6.
(y + 2)(y – 2)
Alternative Solutions:
7.
(2r + s)(2r – s)
Alternative Solutions:
Sumber
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