Problem 77 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each algebraic expression for the given value of the variable. $$-x^{2}-10 x ; x=-1$$

Step-by-Step Solution

Answer

By substituting \(x = -1\) into the expression \(-x^2 - 10x\), we get a result of 9.

Step 1: Substitute the given value

The first step is substitution. Replace each \(x\) in the given expression \(-x^2 - 10x\) with the value \(-1\). Doing this, the expression becomes \(-(-1)^2 -10(-1)\).

Step 2: Simplify the expression

Second step is to simplify the expression. The square of -1 is 1, and the product of -10 and -1 is 10. This makes the expression becomes \(-(1) + 10\) = 9.

Key Concepts

Algebraic Substitution

Algebraic substitution is a key concept in algebra that involves replacing a variable in an expression with a specific value. This is often the first step in evaluating algebraic expressions.
In our example, the expression \( -x^2 - 10x \) is given with the variable \( x = -1 \). Substituting \( -1 \) for \( x \), transforms the expression into \( -(-1)^2 - 10(-1) \).

• Substitute the value: Identify the variable in the expression.
• Replace it with the given number. Be sure to substitute the exact value provided.
• Ensure the entire expression reflects the substitution without leaving any terms in terms of the variable.

This method helps in paving the way for simplification, as it breaks down complex equations into more manageable numbers.

Simplifying Expressions

Simplifying expressions is one of the fundamental skills in algebra. After substitution, the next step is to simplify the components by following the order of operations. This means you should calculate exponents first, followed by multiplication or division, and finally addition or subtraction.
For example, in the transformation from \( -(-1)^2 - 10(-1) \), follow these steps:

• Calculate \((-1)^2 \). The result is 1 because multiplying \(-1 \times -1\) cancels the negatives, giving a positive product.
• Multiply \(-10 \times -1 \). The result is 10 since the negative signs cancel out, turning it into a positive number.
• Lastly, perform the addition and subtraction: \(-(1) + 10 \), which simplifies to \(9\).

Remember, always respect parentheses and signs to avoid errors during this simplification process.

Evaluating Polynomials

Evaluating polynomials involves a calculated substitution and simplification process. A polynomial is an expression made up of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents.
In the expression \(-x^2 - 10x\), we see a polynomial of degree 2, due to the \(x^2\) term. Evaluating such expressions follows a systematic approach:

• Identify the degree and the terms involved. This expression has two terms: \(-x^2 \) and \(-10x\).
• Apply substitution by inputting the given value for \(x\).
• Simplify by handling exponents first and then progressing to other operations.

The key to successfully evaluating polynomials is accurate substitution and methodical simplification, ensuring no step is overlooked or incorrectly executed.