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Chapter 4: Problem 53

Add \(\left(3 x y^{2}+2 x y+5 x^{2} y\right)+\left(2 x y^{2}-4 x y+2 x^{2} y\right)\).

Short Answer

Expert verified
The sum of the given polynomials is \(5 x y^{2} - 2 x y + 7 x^{2} y\).

Step by step solution

01

Identify the like terms

The given polynomials are: \[\left(3 x y^{2}+2 x y+5 x^{2} y\right)+\left(2 x y^{2}-4 x y+2 x^{2} y\right)\] In this expression, the like terms are: - \(3xy^2\) and \(2xy^2\) (both have \(x y^{2}\)) - \(2xy\) and \(-4xy\) (both have \(xy\)) - \(5x^2y\) and \(2x^2y\) (both have \(x^2 y\))
02

Add the coefficients of the like terms

Now, we need to add the coefficients of the like terms: - For \(x y^{2}\) terms: \(3 + 2 = 5\). Thus, the combined term is \(5 x y^{2}\). - For \(x y\) terms: \(2 - 4 = -2\). Thus, the combined term is \(-2 x y\). - For \(x^2 y\) terms: \(5 + 2 = 7\). Thus, the combined term is \(7 x^{2} y\).
03

Write down the final result

Putting the combined terms together, we get: \[5 x y^{2} - 2 x y + 7 x^{2} y\] So, the sum of the given polynomials is \(5 x y^{2} - 2 x y + 7 x^{2} y\).

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Key Concepts

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

Like Terms
When dealing with polynomials, identifying like terms is crucial for performing addition or subtraction. Like terms are terms that have the same variables raised to the same power. Only their coefficients differ.

For example, in the polynomials given:
  • The terms \(3xy^2\) and \(2xy^2\) are like terms because they both contain the variable combination \(xy^2\).
  • The terms \(2xy\) and \(-4xy\) also match because they include the variable combination \(xy\).
  • Lastly, \(5x^2y\) and \(2x^2y\) share the variables \(x^2y\).
Recognizing these similarities helps in combining the terms correctly.

Essentially, like terms make it possible to simplify polynomial expressions efficiently.
Polynomial Coefficients
Coefficients are the numerical factors placed in front of variables in a term. When adding polynomials, we only need to add or subtract these coefficients for like terms.

In the expression:
  • For the terms \(3xy^2\) and \(2xy^2\), their coefficients are 3 and 2, which sum up to 5.
  • The terms \(2xy\) and \(-4xy\) carry coefficients of 2 and -4, resulting in a combined coefficient of -2.
  • Lastly, \(5x^2y\) and \(2x^2y\) have coefficients 5 and 2, which add up to 7.
Understanding coefficients is essential for combining terms correctly. It is the key to correctly managing and simplifying polynomial expressions.
Simplifying Polynomials
Simplifying polynomials involves combining like terms into a more compact expression. Each term should be checked for like terms, and their coefficients should be operated on accordingly.

For instance, in this example:
  • First, like terms were identified: \(xy^2\), \(xy\), and \(x^2y\).
  • The coefficients were then added: 3 + 2, 2 - 4, and 5 + 2.
  • Resulting in a simplified polynomial: \(5xy^2 - 2xy + 7x^2y\).
This simplified form is more straightforward and easier to work with.

Through this process, the polynomial becomes more manageable, which aids in further calculations or evaluations. It's a fundamental skill in algebra, useful for many applications in mathematics.

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Most popular questions from this chapter

Combine and simplify \(1+\\{1 /[1+1 /(1-\mathrm{x})]\\}\).

Simplify \(\quad(5 a x+b y)(2 a x-3 b y)\)

From the sum of \(6 x^{2}+4 x y-8 y^{2}-11\) and \(3 x^{2}-4 y^{2}+8+5 x y\) subtract \(\mathrm{xy}-10-5 \mathrm{x}^{2}+7 \mathrm{y}^{2}\)

Combine into a single fraction in lowest terms. (a) \(\\{6(a+1) /(a+8)\\}-\\{3(a-4)\\} /\\{(a+8)\\}-\\{2(a+5)\\} /\\{(a+8)\\}\) (b) \(\\{(7 x-3 y+6)\\} /(x+y)-\\{2(x-4 y+3)\\} /(x+y)\) (c) \(\\{(5 x+2)\\} /(x-6)-\\{3(x+4)\\} /(x-6)-\\{(x-7)(x-6)\\}\)

(1)What is the difference between a polynomial and a multinomial? (2) What is the degree of a monomial?
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