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Chapter 4: Problem 5

\(\left(12 x^5-3 x^4+2 x-5\right)+\left(8 x^4-3 x^3+4 x+1\right)\)

Short Answer

Expert verified
The result of adding these two polynomials together is \(12 x^5+5 x^4+3 x^3+6 x-4\).

Step by step solution

01

Identify Like Terms

Like terms in these two polynomials are: \(12 x^5\) and none, \(-3 x^4\) and \(8 x^4\), \(2 x\) and \(4 x\), \(-5\) and \(1\). It should be noted that \(3 x^3\) has no like term in the first polynomial, which means it will be included as it is in the solution.
02

Add Like Terms

Adding like terms gives: \(12 x^5 + 0\), \(-3 x^4 + 8 x^4\), \(2 x + 4 x\), \(-5 + 1\), and adding \(3 x^3\) as it is.
03

Simplify the Expression

Simplifying these gives us: \(12 x^5\), \(5 x^4\), \(6 x\), \(-4\), adding \(3 x^3\). So the combined polynomial is \(12 x^5+5 x^4+3 x^3+6 x-4\).

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Key Concepts

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

Like Terms
Understanding like terms is essential for simplifying polynomials. In algebra, like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients.

For example, in the polynomials given in the exercise, the like terms identified are:
  • \(12x^5\) from the first polynomial,
  • \(-3x^4\) and \(8x^4\) from both polynomials,
  • \(2x\) and \(4x\),
  • \(-5\) and \(1\).
Notice that \(3x^3\) doesn't have a matching term in the first polynomial. It simply carries over as is. Combining these like terms helps streamline the polynomial for further calculations.
Polynomial Simplification
Simplifying polynomials involves combining like terms to reduce the expression to a simpler form. This simplification gives us a clearer view of the polynomial's structure and makes it easier to work with.

Here鈥檚 how you do it:
  • Combine the coefficients of like terms.
  • Ensure each term's variable and degree remain unchanged.
In the given problem, we simplified the following:
  • \(-3x^4 + 8x^4 = 5x^4\)
  • \(2x + 4x = 6x\)
  • \(-5 + 1 = -4\)
The resulting polynomial becomes \(12x^5 + 5x^4 + 3x^3 + 6x - 4\). The process helps reduce complexity, making it more convenient for further mathematical operations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra and are essential for communicating mathematical ideas effectively.

In the exercise provided, each part such as \(12x^5\), \(-3x^4\), and so on is a term of a polynomial. Here鈥檚 what you need to know about each part:
  • The numerical part is called the coefficient.
  • The variable part represents an unknown or a set of unknowns.
  • The exponent attached to the variable indicates the degree of the term.
In a polynomial, these expressions are combined using addition or subtraction. Working with algebraic expressions requires identifying these characteristics so you can manipulate them correctly. This understanding fosters better problem-solving skills and allows for easy manipulation of complex expressions.

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Most popular questions from this chapter

Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ 8 m^3-343 $$

Show that the binomial is a factor of the polynomial. Then factor the function completely. $$ h(x)=x^3-x^2-24 x-36 ; x+2 $$

Show that the binomial is a factor of the polynomial. Then factor the function completely. $$ t(x)=x^3-5 x^2-9 x+45 ; x-5 $$

The profit \(P\) (in millions of dollars) for a T-shirt manufacturer can be modeled by \(P=-x^3+4 x^2+x\), where \(x\) is the number (in millions) of T-shirts produced. Currently the company produces 4 million T-shirts and makes a profit of \(\$ 4\) million. What lesser number of T-shirts could the company produce and still make the same profit?

\((3-2 i)+(5+9 i)\)
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