Question

As parts (a) and (b) of Example 3 show, it is sometimes easier to evaluate an expression by simplifying it before substituting, and sometimes easier if you substitute for the variable first. a. Write an expression that is easier to evaluate if you simplify before substituting 12 for \(x\) b. Write an expression that is easier to evaluate if you substitute 12 for \(x\) first.

Short answer

a. An expression that is easier to evaluate by simplifying before substituting is \((x^{2} - 1) / (x - 1)\) \nb. An expression that is easier to evaluate by substituting first is \((x + 3)^2\)

Step-by-step solution

Generate an expression that is easier to evaluate by simplification first

Consider the expression \((x^{2} - 1) / (x - 1)\). If \(x = 12\) is substituted immediately, the division operation makes it complex. However, by simplifying first using difference of squares to rewrite the expression as \((x - 1)(x + 1) / (x - 1)\), the term \((x - 1)\) in the numerator and the denominator cancel each other, simplifying the expression to \(x + 1\). Following this, we can easily substitute \(x = 12\), which gives us 13 as the result.

Generate an expression that is easier to evaluate by substitution first

Consider the expression \((x + 3)^2\). If \(x = 12\) is substituted immediately, we get \((12 + 3)^2 = 15^2 = 225\). This is simpler than first expanding the expression to \(x^2 + 6x + 9\) and then substituting \(x = 12\), which will require more arithmetic operationsvto arrive at the same result.