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Chapter 5: Problem 56

Find \(P(4)\) and \(P(0): P(x)=3 x^{2}-2 x+7\)

Short Answer

Expert verified
P(4)=47, P(0)=7.

Step by step solution

01

Understand the Polynomial

The given polynomial is defined as: \[P(x) = 3x^2 - 2x + 7\].
02

Substitute x = 4

To find \(P(4)\), substitute \(x = 4\) into the polynomial:\[P(4) = 3(4)^2 - 2(4) + 7\].
03

Simplify the Expression for P(4)

Calculate the value step-by-step:\[P(4) = 3(16) - 2(4) + 7\]\[P(4) = 48 - 8 + 7\]\[P(4) = 47\].
04

Substitute x = 0

To find \(P(0)\), substitute \(x = 0\) into the polynomial:\[P(0) = 3(0)^2 - 2(0) + 7\].
05

Simplify the Expression for P(0)

Calculate the value step-by-step:\[P(0) = 0 - 0 + 7\]\[P(0) = 7\].

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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 common technique used in polynomial evaluation. It involves replacing the variable in the polynomial function with a specific value. For example, if we have a polynomial function \(P(x) = 3x^2 - 2x + 7\), and we want to find \(P(4)\), we substitute \(x = 4\). This transforms our polynomial into \(P(4) = 3(4)^2 - 2(4) + 7\). This method simplifies the process of finding the value of the polynomial at specific points.
When substituting, ensure to replace every occurrence of the variable with the given number.
Always double-check your substitution to avoid errors.
Simplifying Expressions
Simplifying expressions means breaking down and combining like terms to find a single value. This step is critical after substituting a value into a polynomial. Let's consider \(P(4) = 3(4)^2 - 2(4) + 7\). First, calculate \(4^2 = 16\), so we have \(P(4) = 3(16) - 2(4) + 7\).
Next, multiply the constants: \(3 \times 16 = 48\) and \(2 \times 4 = 8\). Now, combine them: \(48 - 8 + 7\).
The final step is simple arithmetic: \(48 - 8 = 40\) and then \(40 + 7 = 47\). Hence, \(P(4) = 47\). Similarly, for \(P(0) = 3(0)^2 - 2(0) + 7\), simplifying gives us \(0 - 0 + 7 = 7\).
Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For example, \(P(x) = 3x^2 - 2x + 7\) is a polynomial in one variable \(x\).
Polynomials can have different degrees, which indicate the highest power of the variable present. In our example, the degree is 2, as the highest power of \(x\) is \(x^2\).
These functions can be evaluated at specific points using methods like substitution, and their expressions can be simplified to derive meaningful results.
Understanding these basics helps in manipulating and solving various polynomial-related mathematical problems effectively.

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