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Chapter 6: Problem 44

Simplify each algebraic expression. \(7 x^{2}-10 x^{2}\)

Short Answer

Expert verified
\(-3x^2\)

Step by step solution

01

Identify Like Terms

Look at your expression, which is \(7x^2 - 10x^2\). Notice that both \(7x^2\) and \(-10x^2\) are 'like' terms - they are coefficients of the term 'x' raised to the same power.
02

Combine Like Terms

Now combine these 'like' terms by carrying out the subtraction operation between their coefficients. So combine 7 (the coefficient of the first 'x' term) and -10 (the coefficient of the second 'x' term) to get -3.
03

Write Your Simplified Expression

Now that you've combined your like terms, write your simplified expression. Your new coefficient is -3, so the simplified expression is \(-3x^2\).

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Key Concepts

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

Like Terms
In algebraic expressions, 'like terms' refer to terms that have the same variables raised to the same powers. In the expression \(7x^2 - 10x^2\), both terms are like terms because they share the same variable, \(x\), and the variable is raised to the power of 2 in both terms. Therefore, they can be combined.
Identifying like terms is crucial because it allows us to simplify expressions by combining them. Here are some key points to remember about like terms:
  • They must have identical variable parts.
  • It does not matter if the coefficients (the numbers in front) are different.
  • Only like terms can be combined by adding or subtracting the coefficients.
Looking for like terms first helps streamline the simplification process, making math less complex.
Simplification
Simplification in algebra involves condensing expressions into their simplest form. By simplifying, you make expressions more manageable and easier to interpret. Taking our original expression, \(7x^2 - 10x^2\), the simplification is done by first identifying like terms and then combining them.
Simplifying an expression can involve:
  • Combining like terms.
  • Reducing fractions if they are present.
  • Removing any unnecessary operations or terms.
This process reduces the complexity of problems, making it easier to solve equations or perform other operations in future steps. Simplified expressions are not only easier to read but can also reveal insights or solutions that may be hidden in a more complex form.
Coefficients
Coefficients are the numerical components of algebraic terms. They are the "multiplier" of the variables. For instance, in \(7x^2\), the coefficient is 7, and in \(-10x^2\), the coefficient is -10. Coefficients play a significant role in manipulating algebraic expressions because they determine the magnitude of each term.
When working with coefficients:
  • You perform operations like addition or subtraction on them when combining like terms.
  • The sign of the coefficient (positive or negative) affects the result of operations.
  • Keeping track of coefficients ensures accurate simplification, avoiding errors in calculations.
Understanding coefficients is essential because they give weight to the terms in an expression, directly impacting the expression's value or meaning once simplified.

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Most popular questions from this chapter

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(3 x^{2}+7 x+2\)

A phone plan costs \(\$ 20\) per month for 60 calling minutes Additional time costs \(\$ 0.40\) per minute. The formula $$ C=20+0.40(x-60) $$ gives the monthly cost for this plan, \(C\), for \(x\) calling minutes, where \(x>60\). How many calling minutes are possible for a monthly cost of at least \(\$ 28\) and at most \(\$ 40\) ?

Use FOIL to find the products in Exercises 1-8. \((x+7)(x+2)\)

Solve each equation by the method of your choice. \((x-1)(3 x+2)=-7(x-1)\)

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(x^{2}-x-90\)
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