Using Keywords for Symbolic Operations

When you evaluate an expression with the symbolic equal sign, Mathcad simplifies the result by performing arithmetic and combining like variables.

2x + 5 + 4x − 3 → 6·x + 2

To perform more complex symbolic operations, you can insert a keyword that specifies the operation before the symbolic equal sign. This section explains how to use keywords, and covers the following topics:

Inserting a Keyword

The following example illustrates how to insert a keyword to perform a symbolic operation. Suppose you want to factor the polynomial x2 − 2x + 15. To do so, type the polynomial and insert the keyword "factor" as follows:

  1. Click anywhere in the expression for the polynomial.

  2. Press [Ctrl] [Shift] [.] to insert a placeholder for the keyword, followed by the symbolic equal sign.

    Inserting a placeholder for a keyword

  3. Type the keyword "factor" in the placeholder.

    Typing the keyword

  4. Press [Enter] or click outside the region.

    x2 − 2x − 15 factor → (x + 3)·(x − 5)

Note: As an alternative to pressing [Ctrl] [Shift] [.] and typing "factor," you can select factor from the Symbolic toolbar.

Using Keywords with Variables that Have Assigned Values

When you apply a keyword to an expression containing a variable that has an assigned value, Mathcad first replaces the variable with its value and then applies the keyword. In the preceding example, if you assign the value x:= 7 before applying "factor," you get the following result:

x:= 7

x2 − 2x − 15 factor → 22·5

In this case, Mathcad substitutes 7 for x in the polynomial and evaluates it to get 20, and then factors the integer 20 into prime factors.

If you do not want Mathcad to substitute the assigned value of a variable before applying the keyword, you can either type x:= x to clear the symbolic value of x (see Clearing the Symbolic Value of a Variable), or use the keyword "explicit" to temporarily suppress the value of the variable.

Using Multiple Keywords

Some problems require you to use more than one keyword. For example, to expand the function ex in a Taylor series and then substitute the value x = 2, use the keywords "series" and "substitute" as follows:

  1. Type ex and press [Ctrl] [Shift] [.].
  2. In the placeholder that appears, type the keyword "series."
  3. Press [Enter] to see the result of the keyword "series."

    series for the exponential function

  4. Click in the result to the right of the symbolic sign and press [Ctrl] [Shift] [.]. The result temporarily disappears and a placeholder appears.

    placeholder for second keyword

  5. In the placeholder, type "substitute, x=2" and press [Enter].

    two keywords chained together

The result is the same as what you would get by copying and pasting the result after "series" to a new line, and then applying the "substitute, x = 2" command in the second line.

Stacking Keywords

Another way to combine keywords is stack them vertically before a single symbolic evaluation. To stack the keywords "series" and "substitute" in the preceding example, repeat steps 1 and 2 above, and then do the following:

  1. Press [Ctrl] [Shift] [.] again to create a vertical bar after ex and a placeholder below "series."

    keyword series with placeholder below

  2. In the placeholder, type "substitute, x=2", using the Boolean equals, [Ctrl] [=].
  3. Press [Enter].

    keywords_series_substitute

Note: Stacking keywords does not always return the same result as evaluating them in a sequence. When you want the second keyword to control the behavior of the first keyword, you should stack the keywords. In the following example, the second keyword "assume" controls the behavior of the first keyword "solve" by requiring the variable x to be real.

Solving a polynomial assuming the variable is real

In this case, evaluating the keywords in a sequence would not return the correct result.

Note: When keywords are stacked, their operations are carried out in descending order. In the first example above, Mathcad first expands the function as a series and then substitutes x = 2.

Displaying Keywords

You can choose to hide keywords or the entire left-hand side of an expression. Right click on the keywords or the left-hand expression and choose Hide keywords or Hide left-hand side. Click outside the region to see the new display. Uncheck these options to make the expressions and keywords visible again.

You can also change the display of the symbolic evaluation symbol to an equals sign.

List of Keywords and Their Descriptions

The following lists the keywords and the operations they perform:

Keyword Description
assume Make assumptions about the domain of the variables.
coeffs Return the coefficients of a polynomial.
collect Collect terms containing like powers of a variable.
combine Combine terms in an expression using properties of elementary functions.
confrac Calculate the continued fraction expansion of a number or function.
expand Multiply powers and products from an expression.
explicit Return expressions with the values of variables substituted in place, but without reducing numerical expressions.
factor Factor an expression.
float Return results with available numeric values reduced using floating point calculations to the specified precision.
parfrac Expand a rational expression into a sum of fractions with linear or quadratic denominators.
rectangular Return results involving complex numbers separated into real and imaginary parts.
rewrite Rewrite expressions in terms of elementary functions.
series Expand a function or expression in a Taylor or Laurent series around 0.
simplify Algebraically simplify or evaluate an expression.
solve Solve an equation symbolically.
substitute Replace all occurrences of a variable with another variable, an expression, or a number.
transforms:
fourier, laplace,
and z
Evaluate the transform or inverse transform of a function.

Modifiers

Modifiers are special terms that you can use to modify the behavior of keywords. For example, you can use the modifier "real" after the keyword "assume" to require that a variable be a real number:

assume, x = real

To see which modifiers you can use with a specific keyword, see the Help page for the keyword. For a complete list of the modifiers, see Modifiers.

You can use the Modifier Toolbar to insert modifiers.

Tutorial - Using Symbolic Calculation

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