Evaluation: Evaluating
Expressions, Polynomials, and Functions
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Sections: Evaluating Expressions and Polynomials, Evaluating Functions

"Evaluation" mostly means "simplifying an expression down to a single numerical value". Sometimes you will be given a numerical expression, where all you have to do is simplify; that is more of an order-of-operations kind of question. In this lesson, I'll concentrate on the "plug and chug" aspect of evaluation: plugging in values for variables, and "chugging" my way to the simplified answer.

Usually the only hard part in evaluation is in keeping track of the minus signs. I would strongly recommend that you use parentheses liberally, especially when you're just getting started.

• Evaluate a2b for a = –2, b = 3, c = –4, and d = 4.

(–2)2(3) = (4)(3) = 12

• Evaluate a cd for a = –2, b = 3, c = –4, and d = 4.
• (–2) – (–4)(4) = –2 – (–16) = –2 + 16 = 16 – 2 = 14

• Evaluate (b + d)2 for a = –2, b = 3, c = –4, and d = 4.

I must take care not to try to "distribute" the exponent through the parentheses. Exponents do NOT distribute over addition! I should never try to say that (b + d)2 is the same as b2 + d2! They are NOT the same thing! I must evaluate the expression as it stands:

( (3) + (4) )2 = ( 7 )2 = 49

• Evaluate b2 + d2 for a = –2, b = 3, c = –4, and d = 4.
• (3)2 + (4)2 = 9 + 16 = 25

Notice that this does not match the answer to the previous evaluation, pointing out again that exponents do not "distribute" the way multiplication does.

• Evaluate bc3ad for a = –2, b = 3, c = –4, and d = 4.
• (3)(–4)3 – (–2)(4) = (3)(–64) – (–8) = –192 + 8 = –184

The most common "expression" you'll likely need to evaluate will be polynomials. To evaluate, you take the polynomial and plug in a value for x.

• Evaluate x4 + 3x3x2 + 6 for x = –3.
• (–3)4 + 3(–3)3 – (–3)2 + 6
= 81 + 3(–27) – (9) + 6
= 81 – 81 – 9 + 6
= –3

• Evaluate 3x2 – 12x + 4 for x = –2.
• 3(–2)2 – 12(–2) + 4 = 3(4) + 24 + 4 = 12 + 24 + 4 = 40

• Evaluate y = 4x – 3 at x = –1.
• y = 4(–1) – 3 = –4 – 3 = –7

Note: This means that the point (–1, –7) is on the line y = 4x – 3.