Polynomials and Value of an Expression

Polynomials and Value of an Expression

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As the values of the variables in the expression are changed, the value of the expression itself will change (in general). For example, in the expression \(3x + 2,\) as the value of x is increased, the value of the expression will change (in fact it will increase). Thus, the value of an expression is in general variable because the expression is composed of terms which themselves have variable values.

A special class of expressions is the class of constant expressions – expressions whose values do not change, perhaps because they have no variable in them. For example, the expressions

\[\begin{array}{l}{2^3}\\\pi + \sqrt 2 + 1\end{array}\]

are constant expressions – their values are fixed. But in general, the value of an expression is variable.

To evaluate the value of an expression, we need to know what values its variables have taken. Different variable values will generate different values for the expression. Consider the expression \(3x + 2\). Let us denote this expression by E. What value will E take when x is equal to 2? Well, we simply substitute x as 2 in E:

\[3\left( 2 \right) + 2 = 6 + 2 = 8\]

Thus, the value of E when x equals 2 is 8. We will write this fact as follows:

\[E\left( 2 \right) = 8\]

And we will read it as follows: The value of E at 2 is 8. Verify the following:

\[\begin{array}{l}E\left( 1 \right) = 5, & E\left( \pi \right) = 3\pi + 2\\E\left( 0 \right) = 2\end{array}\]

Now, consider the expression \(3x + 5y\). Let us denote this expression by F. What value will F take when x is equal to 3 and y is equal to \( - 1\)? We substitute x as 3 and y as \( - 1\) in F:

\[3\left( 3 \right) + 5\left( { - 1} \right) = 9 + \left( { - 5} \right) = 4\]

Thus, the value of F at the given x and y values is 4. We will write this fact as follows:

\[F\left( {3,\,\, - 1} \right) = 4\]

On the left hand side above, the first number, 3, denotes the value that x takes, and the second number, \( - 1\), denotes the value that y takes. Now, verify the following assertions:

\[\begin{array}{l}F\left( {0,\,\,0} \right) = 0, & & F\left( {1,\,\,8} \right) = 43\\F\left( {2,\,\,\sqrt 2 } \right) = 6 + 5\sqrt 2 \end{array}\]

Thus, to evaluate the value of an expression at a given set of variable values, simply substitute the values of the variables into the expression.

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