Problem 22 Add or subtract the monomials. ... [FREE SOLUTION]

Chapter 6: Problem 22

Add or subtract the monomials. $$5 u^{2}+4 v^{2}-6 u^{2}$$

Short Answer

Expert verified

-u^2 + 4v^2

Step by step solution

Identify Like Terms

Look at the given monomials and identify which ones are like terms. Monomials with the same variable and exponent can be added or subtracted.

Group Like Terms

Group the like terms together. In this case, group the terms with the variable $u^2$ together: $5u^2 - 6u^2$.

Perform the Operation on Like Terms

Add or subtract the coefficients of the grouped like terms: $5u^2 - 6u^2 = (5 - 6)u^2 = -1u^2 = -u^2$.

Combine Remaining Terms

Combine the result with the non-like term. Since $4v^2$ cannot be combined with $u^2$, it remains the same. The final expression is: $-u^2 + 4v^2$.

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Key Concepts

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

like terms

In mathematics, especially in algebra, it's essential to understand the concept of 'like terms'.
Like terms are terms that have the same variables raised to the same power. For example, in the expression given:$5u^2 + 4v^2 - 6u^2$, the terms $5u^2$ and $ -6u^2$ are 'like terms' because both have the same variable 'u' raised to the power of 2.
They can be combined through addition or subtraction.
When you identify like terms, you're halfway to simplifying an expression. Knowing how to group and combine them makes algebra more straightforward and manageable.

polynomials

A polynomial is an expression made up of variables, coefficients, and constants, that involves adding, subtracting, and multiplying, but not dividing by a variable.
Polynomials are composed of terms, which are monomials like the ones in the exercise: $5u^2, 4v^2, -6u^2$. Each term can be a constant (like 4), a variable with a coefficient (like $5u^2$), or a combination of both.
The degree of a polynomial is determined by the highest power of the variable in the expression. Understanding and working with polynomials is a fundamental part of algebra. It helps in solving equations, understanding functions, and much more.

coefficients

Coefficients are the numerical part of terms in algebraic expressions.
They are the numbers in front of the variables. For example, in the term $ 5u^2$, 5 is the coefficient, and in the term $ -6u^2$, -6 is the coefficient.
When adding or subtracting polynomials, you only combine like terms by operating on their coefficients. The coefficients tell you how many times to multiply the variable they precede.
In our original exercise: $ 5u^2 + 4v^2 - 6u^2 $, we add or subtract the coefficients of the like terms to simplify the polynomial. Remember, coefficients play a crucial role in determining the value and behavior of a polynomial.

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91影视

Add or subtract the monomials. $$5 u^{2}+4 v^{2}-6 u^{2}$$

Short Answer

Step by step solution

Identify Like Terms

Group Like Terms

Perform the Operation on Like Terms

Combine Remaining Terms

Key Concepts

like terms

polynomials

coefficients

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