Problem 8
Question
Add. $$ (-5 x 2-1+x)+(-x+7 x 2-9) $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(2x^2 - 2x - 8\).
1Step 1: Identify Like Terms
Look at each polynomial expression and identify like terms. `(-5x^2 - x + 1)` contains terms with \(x^2\), \(x\), and a constant. Similarly, `(-x + 7x^2 - 9)` also contains terms with \(x^2\), \(x\), and a constant.
2Step 2: Reorganize Terms
Rearrange the expression so that like terms are next to each other: \((-5x^2 + 7x^2) + (-x - x) + (1 - 9)\).
3Step 3: Combine Like Terms
Add the coefficients of like terms together: \(-5x^2 + 7x^2 = 2x^2\), \(-x - x = -2x\), and \(1 - 9 = -8\).
4Step 4: Form the Simplified Expression
Write the expression with combined like terms: \(2x^2 - 2x - 8\).
Key Concepts
Like TermsSimplifying ExpressionsAlgebraic Expressions
Like Terms
In algebra, like terms are terms that contain the same variables raised to the same power. For example, in the expressions
When adding polynomials, always look for like terms and group them together. This simplifies the expression, making it easier to work with and understand.
- \(-5x^2\) and \(7x^2\) are like terms because they both contain the variable \(x\) raised to the power of 2.
- Similarly, \(-x\) and \(-x\) are also like terms since they both involve the variable \(x\) raised to the first power.
When adding polynomials, always look for like terms and group them together. This simplifies the expression, making it easier to work with and understand.
Simplifying Expressions
Simplifying algebraic expressions involves reducing them to their simplest form. This process is achieved by combining like terms.
After identifying and grouping like terms, the next step is to add or subtract their coefficients. Coefficients are the numerical parts of the terms.
This makes the expression clearer and more manageable for any subsequent operations or evaluations.
After identifying and grouping like terms, the next step is to add or subtract their coefficients. Coefficients are the numerical parts of the terms.
- For example, when you have \(-5x^2 + 7x^2\), add their coefficients (-5 and 7) to get \(2x^2\).
- Similarly, for \(-x - x\), think of \(-x\) as \(-1x\) : -1 - 1 = -2, resulting in \(-2x\).
This makes the expression clearer and more manageable for any subsequent operations or evaluations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols. They can represent real-world situations in mathematical terms.
In our example, the original expression \((-5x^2 - x + 1) + (-x + 7x^2 - 9)\) is an algebraic expression comprising two polynomials.
Working with algebraic expressions often involves manipulating them through operations such as addition, subtraction, multiplication, or simplification.
Using standard rules, such as combining like terms, allows us to simplify and operate on these expressions efficiently.
In our example, the original expression \((-5x^2 - x + 1) + (-x + 7x^2 - 9)\) is an algebraic expression comprising two polynomials.
Working with algebraic expressions often involves manipulating them through operations such as addition, subtraction, multiplication, or simplification.
Using standard rules, such as combining like terms, allows us to simplify and operate on these expressions efficiently.
- This leads to a simplified form: \(2x^2 - 2x - 8\), which is easier to interpret and use.