“The product of two linear expressions is quadratic but the sum of two linear expressions is still linear”.  Do you agree with the statement? Show an example to support your answer.

Expert Answers

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The first part is true but the second part isn't. Let `ax+b` and `cx+d` be linear expressions. Their product is `(ac)x^2+(ad+bc)x+db.` Now neither `a` or `c` is zero, or else at least one of `ax+b` and `cx+d` would be constant, not linear. Therefore, `ac!=0,` and the product is quadratic.

The second part isn't true because `x+1` and `-x+1` are both linear expressions, but their sum is `2,` which is constant but not linear.


BUT, I couldn't find a consistent definition for "linear expression" online, so I used the definition in the link below. If something like `2` is considered a linear expression (since the graph of `y=2` is a line, after all), then the sum of two linear expressions would always be linear, but the product of the linear expressions `2` and `3` would almost certainly not be considered "quadratic", so the answer is completely switched. You'll have to check your book or your notes to see exactly what is meant by linear expression. At least in either case, the statement is false.

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