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Chapter 4: Problem 45

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

Short Answer

Expert verified
When x = 2, the polynomial value is 19. When x = -1, the polynomial value is -2.

Step by step solution

01

Substitute x = 2 into the polynomial

To find the value of the polynomial when x = 2, substitute 2 for x in the given polynomial: 2(2)^2 + 5(2) + 1.
02

Calculate the polynomial for x = 2

First, calculate the square of 2: 2^2 = 4. Substituting back in: 2(4) + 5(2) + 1. Next, perform the multiplications: 2(4) = 8 and 5(2) = 10. Finally, add all terms together: 8 + 10 + 1 = 19. Hence, when x = 2, the value of the polynomial is 19.
03

Substitute x = -1 into the polynomial

To find the value of the polynomial when x = -1, substitute -1 for x in the given polynomial: 2(-1)^2 + 5(-1) + 1.
04

Calculate the polynomial for x = -1

First, calculate the square of -1: (-1)^2 = 1. Substituting back in: 2(1) + 5(-1) + 1. Next, perform the multiplications: 2(1) = 2 and 5(-1) = -5. Finally, add all terms together: 2 - 5 + 1 = -2. Hence, when x = -1, the value of the polynomial is -2.

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Key Concepts

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

Substitution in Polynomials
To evaluate a polynomial at a specific value, we use a technique called substitution. This means we replace the variable in the polynomial with a given number. By doing this, we can turn the polynomial into a simple arithmetic problem. For example, let's consider the polynomial expression given: \[2x^2 + 5x + 1\]. If we need to find the value of this polynomial when \(x = 2\), we substitute 2 for \(x\).
So, our expression becomes: \(2(2)^2 + 5(2) + 1\). It's important to follow the order of operations:
  • First, calculate the exponent: \(2^2 = 4\)
  • Substitute back in: \(2(4) + 5(2) + 1\)
  • Next, perform the multiplication: \(2(4) = 8\) and \(5(2) = 10\)
  • Finally, add all the terms together: \(8 + 10 + 1 = 19\).
By substituting different values for \(x\), we can find the corresponding value of the polynomial.
Polynomial Calculations
Calculating the value of polynomials involves a few key steps. You鈥檒l need to understand the order of operations (also known as PEMDAS): Parentheses, Exponents, Multiplication and Division (left to right), Addition, and Subtraction (left to right). Let's break this down using our example polynomial. For \(x = -1\):
  • First, substitute \(-1\) for \(x\): \(2(-1)^2 + 5(-1) + 1\)
  • Calculate the exponent: \((-1)^2 = 1\)
  • Substitute back in: \(2(1) + 5(-1) + 1\)
  • Next, do the multiplication: \(2(1) = 2\) and \(5(-1) = -5\)
  • Finally, add all terms together: \(2 + -5 + 1 = -2\)
The key is to take each operation one step at a time, ensuring accuracy at every turn. Performing the calculations correctly is vital for obtaining the correct value of the polynomial.
Evaluating Expressions
Evaluating expressions involves finding the value of an expression for specific values of its variables. For polynomials, this means substituting a value for each variable and performing the arithmetic operations. Let's revisit the original polynomial: \[2x^2 + 5x + 1\]. When you are given a value for \(x\), the goal is to substitute it into the polynomial and simplify the expression. Here are some steps:
  • Identify the polynomial and its variable.
  • Substitute the given value into the polynomial for the variable.
  • Follow the order of operations to simplify the expression.
  • Combine like terms to get the final value.
By applying these steps methodically, you'll be able to evaluate any polynomial expression efficiently and accurately.

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Most popular questions from this chapter

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 78\. \(\frac{0.000015(42,000,000)}{0.000009(0.000005)}\)

Graph each equation by completing the table of values. $$ \begin{aligned} &y=2 x^{2}-1\\\ &\begin{array}{c|c} \hline x & y \\ \hline-2 & \\ \hline-1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \end{array} \end{aligned} $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 103 \times 97 $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (m+6)^{2} $$

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 30 \frac{1}{3} \times 29 \frac{2}{3} $$
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