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Difference of two squares and the square of a binomial

1. Give an example of a single polynomial with degree 3, whose value is 4 when the variable is equal to 1.

2. Give an example of a single polynomial with degree 3, whose value is 29 when the variable is equal to 2.

3. Find a polynomial with degree 3 whose values are 4 and 29 when the variable is equal to 1 and 2, respectively. How many polynomials that satisfy these constraints are there? Note that in this part, you need to come up with a single polynomial that satisfies these two constraints simultaneously.

4. State the identities of the difference of two squares and the square of a binomial. Why do you think they are important? give at least one example from each identity.

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Please see the attached file.

Give an example of a single polynomial with degree 3, whose value is 4 when the variable is equal to 1.
A general form of single polynomial with degree 3:
Y = ax3 + bx2 + cx + d
Here we need to substitute x = 1. Thus
4 = a + b + c + d
As you can see, there are many solutions to this problem. So we will have to guess for one, say a = b = c = d = 1.
Thus we have the function required:
Y = x3 + x2 + x + 1

Give an example of a single polynomial with degree 3, whose value is ...

Solution Summary

The solution explains in detail the relevant concepts and then provides step-by-step problem-solving instructions in a 4-page Word document.

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