Problem 98 $$\text { If } G(c)=3 c^{4}+c^{2... [FREE SOLUTION]

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Chapter 6: Problem 98

$$\text { If } G(c)=3 c^{4}+c^{2}-9 c-4, \text { find }$$ a) $G(0)$ b) $G(-1)$

Short Answer

Expert verified

a) $G(0) = -4$ b) $G(-1) = 9$

Step by step solution

Identify the Function

We are given the function G(c) = 3c^4 + c^2 - 9c - 4. We need to find the value of G(c) when c is 0 and when c is -1.

Evaluate G(0)

To find the value of G(c) when c = 0, we need to replace "c" in the function with 0: G(0) = 3(0)^4 + (0)^2 - 9(0) - 4 \\ G(0) = 3(0) + 0 - 0 - 4 \\ G(0) = -4

Evaluate G(-1)

To find the value of G(c) when c = -1, we need to replace "c" in the function with -1: G(-1) = 3(-1)^4 + (-1)^2 - 9(-1) - 4 \\ G(-1) = 3(1) + 1 + 9 - 4 \\ G(-1) = 3 + 1 + 9 - 4 \\ G(-1) = 9 So the values for G(c) are: a) G(0) = -4 b) G(-1) = 9

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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 powerful tool for evaluating algebraic expressions, including polynomial functions. Essentially, it involves replacing the variable in the expression with a specific value. This approach allows us to find the output of a function for a given input.When we apply the substitution method:

Identify the variable in the given function.
Replace the variable with the specific value provided in the exercise.
Simplify the expression using basic arithmetic operations.

For instance, if we have a polynomial function like $ G(c) = 3c^4 + c^2 - 9c - 4 $, and we need to evaluate $ G(0) $, we simply replace every "c" in the equation with 0 and simplify the resulting expression. This direct replacement makes it straightforward to solve for particular cases, ensuring a clear path to the solution.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They form the basis for describing relationships in mathematics and can take various forms, including polynomial functions.An algebraic expression like $ 3c^4 + c^2 - 9c - 4 $ is composed of several terms:

Terms: Each part of the expression separated by a plus or minus sign (e.g., $3c^4$, $c^2$).
Coefficients: The numerical part of terms that multiply variables (e.g., 3 in $3c^4$).
Variables: Symbols that stand in for unknown values (e.g., $c$).
Constants: Numbers without variables (e.g., -4).

Understanding these components is crucial as it helps us manipulate expressions. Through operations such as addition, subtraction, and substitution, we can find the value of expressions when given specific variable inputs.

Polynomial Functions

Polynomial functions are a specific type of algebraic expression made up of sums and differences of terms called monomials. Each monomial consists of a constant multiplied by a variable raised to a non-negative whole number power.In a polynomial function such as $ G(c) = 3c^4 + c^2 - 9c - 4 $, we can identify the following:

Degree: This is the highest power of the variable, which in this case is 4 from the term $3c^4$.
Leading Coefficient: The coefficient of the term with the highest degree (3 in $3c^4$).
Constant Term: The term without the variable, here it is -4.

Polynomial functions can describe a wide array of phenomena and are incredibly useful in calculus, physics, and engineering. The process of evaluating these functions helps us understand how changes in the variable affect the overall expression. By mastering polynomial functions, students can tackle more complex mathematical models.

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91影视

$$\text { If } G(c)=3 c^{4}+c^{2}-9 c-4, \text { find }$$ a) \(G(0)\) b) \(G(-1)\)

Short Answer

Step by step solution

Identify the Function

Evaluate G(0)

Evaluate G(-1)

Key Concepts

Substitution Method

Algebraic Expressions

Polynomial Functions

One App. One Place for Learning.

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