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Chapter 5: Problem 40

Find the value of each polynomial for }(a) x=2 \text { and }(b) x=-1 $$ -2 x^{2}+5 x-1 $$

Short Answer

Expert verified
For x=2, the value is 1 . For x=-1, the value is -8.

Step by step solution

01

Understand the Polynomial

The given polynomial is -2x^2 + 5x - 1. We need to find its value at x=2 and x=-1.
02

Substitute x=2 into the Polynomial

Replace x with 2 in the polynomial: -2(2)^2 + 5(2) - 1.
03

Simplify the Expression for x=2

Calculate the powers and multiplications: -2(4) + 5(2) - 1. Then simplify further: -8 + 10 - 1 = 1.
04

Substitute x=-1 into the Polynomial

Replace x with -1 in the polynomial: -2(-1)^2 + 5(-1) - 1.
05

Simplify the Expression for x=-1

Calculate the powers and multiplications: -2(1) + 5(-1) - 1. Then simplify further: -2 - 5 - 1 = -8.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
The substitution method is a key technique in algebra for evaluating polynomials and other expressions. It involves replacing a variable with a given value to simplify the problem.
Here's how it works:
Let's take the polynomial -2x^2 + 5x - 1 and evaluate it for specific values of x.
1. Start with the given polynomial.
2. Substitute the value of x into the polynomial. For example, if x = 2, replace x with 2.
3. Simplify the resulting expression.
This method gives us the value of the polynomial at specific points.Understanding substitution can make solving polynomials more intuitive and straightforward.
simplifying expressions
Simplifying expressions is all about reducing them to their simplest form. This makes calculations easier and results more apparent.
Consider the polynomial again: -2x^2 + 5x - 1. After substituting a value, like x = 2, you get: -2(2)^2 + 5(2) - 1.
To simplify:
* First, solve the power: 2^2 = 4.
* Then multiply: -2 * 4 = -8 and 5 * 2 = 10.
* Finally, combine all terms: -8 + 10 - 1 = 1.
Following these steps helps in breaking down complex expressions into simple calculations.
powers and multiplications
Understanding powers and multiplications is crucial for handling polynomials. Here's a brief overview:
* **Powers**: When a number is raised to an exponent, it multiplies by itself that many times. For instance, 2^2 means 2 * 2 = 4.
* **Multiplications**: After calculating the power, you then multiply by coefficients in the polynomial. For example, -2(2)^2 becomes -2 * 4 = -8.
The final simplification combines all such results, giving us the polynomial's value.
In the exercise, remembering these rules helps to break down and solve each term correctly.

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Most popular questions from this chapter

Subtract. \(5 t^{2}+2 t-6\) \(5 t^{2}-3 t-9\) \(\hline\)

Simplify by writing each expression with positive exponents. Assume that all variables represent nonzero real numbers. See Example \(5 .\) $$ \frac{\left(m^{7} n\right)^{-2}}{m^{-4} n^{3}} $$

Evaluate each expression for \(x=3\) $$ 2 x^{2}-3 x+10 $$

Multiply. $$ (2 a)(-5 a b) $$

Perform each division. $$ \frac{-4 x+3 x^{3}+2}{x-1} $$
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