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Chapter 4: Problem 6

\(\left(8 x^4+2 x^2-1\right)+\left(3 x^3-5 x^2+7 x+1\right)\)

Short Answer

Expert verified
The sum of the polynomials is \(8 x^4+3 x^3-3 x^2+7 x\).

Step by step solution

01

Arrange by Degree

First, arrange the terms in descending order of their degree in both polynomials. The first polynomial becomes \(8 x^4+2 x^2-1\) and the second is \(3 x^3-5 x^2+7 x+1\).
02

Add Like Terms

Add the terms with the same degree from both polynomials: The x^4 terms are: \(8 x^4\), the x^3 terms are: \(3 x^3\), the x^2 terms are: \(2 x^2-5 x^2\), the x terms are: \(7 x\) and the constant terms are: \(-1+1\).
03

Simplify

Now, combine like terms together to simplify the expression. So, our simplified polynomial would be: \(8 x^4+3 x^3-3 x^2+7 x\).

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Key Concepts

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

Polynomials
Understanding polynomials is fundamental in algebra. A polynomial is a mathematical expression involving a sum of powers in one or more variables, with coefficients. Each term in a polynomial is usually composed of:
  • A coefficient (which is a number),
  • A variable (like x),
  • An exponent (which tells how many times the variable is multiplied by itself).
For example, in the term \(8x^4\), 8 is the coefficient, \(x\) is the variable, and 4 is the exponent. In the exercise you looked at, there are two polynomials: \(8x^4 + 2x^2 - 1\) and \(3x^3 - 5x^2 + 7x + 1\). Each of these expressions has its own set of terms with different powers of \(x\). By working with polynomials, you learn to manipulate these terms to perform operations like addition, subtraction, and more.
Like Terms
"Like terms" are terms in a polynomial that have the same variable raised to the same power. Being able to identify like terms is key to adding or subtracting polynomials effectively.
  • Like terms have identical variables and exponents, but can have different coefficients. For example: \(3x^2\) and \(-5x^2\) are like terms because both involve \(x^2\).
  • In contrast, \(3x\) and \(3x^2\) are not like terms because the powers of x are different.
The step-by-step solution highlights how to group like terms from the given polynomials. Doing so makes it straightforward to add these terms together, like \(2x^2 - 5x^2\), which results in \(-3x^2\). Identifying and operating on like terms simplifies algebraic expressions considerably.
Simplifying Expressions
Simplifying expressions helps to make polynomials more manageable and easier to understand. This process involves combining all like terms and reducing the expression to its simplest form.
  • The key is to first ensure all like terms are correctly added or subtracted. For example, simplify \((-1 + 1)\) to 0.
  • Arrange the resulting polynomial by decreasing order of the degree of the terms, from the highest exponent to the lowest.
In the exercise, once like terms were combined, the expression \(8x^4 + 2x^2 - 1 + 3x^3 - 5x^2 + 7x + 1\) was successfully simplified to \(8x^4 + 3x^3 - 3x^2 + 7x\). Simplifying allows for easier handling of polynomials in other operations such as multiplication or division.

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Most popular questions from this chapter

MODELING WITH MATHEMATICS During a recent period of time, the numbers (in thousands) of males \(M\) and females \(F\) that attend degree-granting institutions in the United States can be modeled by $$ \begin{aligned} &M=19.7 t^2+310.5 t+7539.6 \\ &F=28 t^2+368 t+10127.8 \end{aligned} $$ where \(t\) is time in years. Write a polynomial to model the total number of people attending degree- granting institutions. Interpret its constant term.

MODELING WITH MATHEMATICS A farmer plants a garden that contains corn and pumpkins. The total area (in square feet) of the garden is modeled by the expression \(2 x^2+5 x+4\). The area of the corn is modeled by the expression \(x^2-3 x+2\). Write an expression that models the area of the pumpkins.

COMPARING METHODS Find the product of the expression \(\left(a^2+4 b^2\right)^2\left(3 a^2-b^2\right)^2\) using two different methods. Which method do you prefer? Explain.

\(f(x)=(2 x-3)^3 ; g(x)=8 x^3-36 x^2+54 x-27\)

Show that the binomial is a factor of the polynomial. Then factor the function completely. $$ s(x)=x^4+4 x^3-64 x-256 ; x+4 $$
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