Problem 59 $f(x)=(-x-3)^4$; $$ g(x)=x... [FREE SOLUTION]

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BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Ron Larson, Laurie Boswell

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 4: Problem 59

$f(x)=(-x-3)^4$; $$ g(x)=x^4+12 x^3+54 x^2+108 x+80 $$

Short Answer

Expert verified

The function $f(x)$ is almost identical to $g(x)$ with the exception of the constant term - $f(x)$ results in 81 as the constant term, whereas the constant term in $g(x)$ is 80. So, $f(x)$ is not equal to $g(x)$.

Step by step solution

Expand $f(x)$

To begin, expand $f(x)$ using the binomial theorem. $f(x)$ expands to: \[ (-x)^4+4(-x)^3*(-3)+6(-x)^2*(-3)^2+4(-x)*(-3)^3+(-3)^4 \] This simplifies to: \[ x^4+12x^3+54x^2+108x+81 \]

Comparison

Compare this simplified form of $f(x)$ with $g(x)$: we can see that the coefficients and powers of $x$ in the expanded $f(x)$ match with $g(x)$, so they are equivalent with the exception of the constant term.

Examine the constant term

In $f(x)$, the constant term is 81 while in $g(x)$, the constant term is 80. This shows that $f(x)$ and $g(x)$ are not entirely equal, the difference being in the constant term.

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Key Concepts

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

Binomial Theorem

The Binomial Theorem is a powerful tool in mathematics used for expanding expressions that are raised to a power. It provides a straightforward method to handle expressions of the form $ (a + b)^n $. This theorem allows for the systematic expansion of these expressions without directly multiplying them out, which can be particularly tedious when dealing with high powers. In our exercise, the binomial theorem was used to expand $ f(x) = (-x-3)^4 $.
For binomial expansions:

The expansion consists of terms involving the binomial coefficients, which are derived from Pascal's triangle.
The general term is given by $ \binom{n}{k} a^{n-k}b^k $, where $ \binom{n}{k} $ is a binomial coefficient.

By applying the binomial theorem, one can expand polynomial expressions efficiently into a sum of terms, each consisting of products of the variables and their coefficients.

Function Expansion

Function expansion is the process of expressing a complex function in terms of a simpler, more expanded form. This expanded form can make it easier to analyze and understand the behavior of the function. In the case of polynomial functions, expanding allows us to see each term of the polynomial in its entirety. For $ f(x) = (-x-3)^4 $, expanding the function using the binomial theorem resulted in the expression:\[ x^4+12x^3+54x^2+108x+81 \].
This expansion:

Reveals the function's polynomial nature, represented by distinct powers of $ x $.
Exposes the coefficients of each term, which play significant roles in determining the curve's shape when plotted on a graph.
Discloses the constant term at the end of the expanded expression.

Understanding the expanded form is key to comparing different functions, as it makes the contributions of individual terms more explicit.

Constant Term

The constant term in a polynomial function is the term that does not contain the variable $ x $. It represents the value of the function when $ x = 0 $. In the context of the exercise, comparing the expanded form of $ f(x) = (-x-3)^4 $ with $ g(x) = x^4 + 12x^3 + 54x^2 + 108x + 80 $, we observe a discrepancy in the constant term. For $ f(x) $, the constant term was calculated as 81, while for $ g(x) $, it was given as 80.
The constant term:

Is crucial for determining the y-intercept of a polynomial function on a graph.
Can affect the overall equality between two polynomial functions, as seen in the exercise.

Even a small alteration in the constant term can change the behavior and intersection of polynomial graphs on a plane, underlining its importance in polynomial expressions.

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91影视

\(f(x)=(-x-3)^4\); $$ g(x)=x^4+12 x^3+54 x^2+108 x+80 $$

Short Answer

Step by step solution

Expand \(f(x)\)

Comparison

Examine the constant term

Key Concepts

Binomial Theorem

Function Expansion

Constant Term

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