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Difficulty: MediumEvaluating Algebraic Expressions

Evaluate the algebraic expression P=4v3−3uv+2u2P = 4v^3 - 3uv + 2u^2 using the assigned values u=−4u = -4 and v=−2v = -2. Fill in the missing numerical values for the intermediate calculation step and the final evaluated result.

Answer:The calculated value for the isolated term 4v34v^3 is 【-32】, which makes the final evaluated total for the entire expression PP equal to 【-24】.

Answer

The first term evaluates to -32, and the entire expression evaluates to -24.
The expression requires careful substitution of u=−4u = -4 and v=−2v = -2. Following the order of operations (PEMDAS), we first calculate the exponents: (−2)3=−8(-2)^3 = -8 and (−4)2=16(-4)^2 = 16. Next, we perform the multiplication for each term: 4(−8)=−324(-8) = -32, −3(−4)(−2)=−24-3(-4)(-2) = -24, and 2(16)=322(16) = 32. Finally, combining these intermediate values gives −32−24+32=−24-32 - 24 + 32 = -24.

Step-by-Step Solution

1
Substitute the value of v into the first term 4v34v^3 and evaluate according to the order of operations.
4(−2)3=4(−8)=−324(-2)^3 = 4(-8) = -32
Exponents must be evaluated before multiplication. The cube of a negative number remains negative.
2
Substitute the values of u and v into the second term −3uv-3uv and evaluate.
−3(−4)(−2)=−3(8)=−24-3(-4)(-2) = -3(8) = -24
Multiplying two negative numbers yields a positive product, which is then multiplied by the negative coefficient.
3
Substitute the value of u into the third term 2u22u^2 and evaluate.
2(−4)2=2(16)=322(-4)^2 = 2(16) = 32
The square of a negative number is positive. This must be calculated before multiplying by the coefficient.
4
Combine the evaluated terms to find the final total for P.
−32−24+32=−24-32 - 24 + 32 = -24
Adding and subtracting the values from left to right yields the final result.

Key Concept

Evaluating multi-variable algebraic expressions by correctly substituting negative integers and rigorously applying the order of operations.
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Topics

Evaluating Algebraic Expressions

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