Problem 69
Question
Perform the operation and write the result in standard form. \((9 x-4)+\left(2 x^{2}-x+15\right)\)
Step-by-Step Solution
Verified Answer
The solution to the given operation in standard form is \(2x^{2}+8x+11\)
1Step 1 - Identify like terms
Like terms are terms whose variables are the same. In the given expression \((9 x-4)+\left(2 x^{2}-x+15\right)\), \(9x\) and \(-x\) are like terms. Realize there is no term to pair with \(2x^{2}\) meanwhile, -4 pairs with 15.
2Step 2 - Combine like terms
The like terms can be added or subtracted using the operation in front of each term. So, add \(9x\) and \(-x\) to get \(8x\). Also, add \(-4\) and 15 to get 11.
3Step 3 - Write down the final expression
Writing the whole polynomial in descending power order will give \(2x^{2}+8x+11\)
Key Concepts
Combining Like TermsStandard Form of PolynomialsPolynomial Addition
Combining Like Terms
When working with polynomials, a crucial skill is combining like terms. Like terms are terms in a polynomial that have the exact same variable and exponent. This means you can add or subtract these terms because they share the same "type" or "kind" of variable.
To identify like terms, look at each part of the polynomial and notice which terms share variables and exponents. For instance, in the expression \((9x - 4) + (2x^2 - x + 15)\), the terms \(9x\) and \(-x\) are like terms because they both contain the variable \(x\) to the first power, even though they have different coefficients.
An important tip: Always ensure you're only combining terms with the same variable and exponent. If there's no matching term with a particular variable, like \(2x^2\) in our example, simply leave it alone in the computation.
To identify like terms, look at each part of the polynomial and notice which terms share variables and exponents. For instance, in the expression \((9x - 4) + (2x^2 - x + 15)\), the terms \(9x\) and \(-x\) are like terms because they both contain the variable \(x\) to the first power, even though they have different coefficients.
An important tip: Always ensure you're only combining terms with the same variable and exponent. If there's no matching term with a particular variable, like \(2x^2\) in our example, simply leave it alone in the computation.
Standard Form of Polynomials
A polynomial is in standard form when its terms are ordered by degree from highest to lowest. This means you place the term with the highest power first, followed by terms of lesser power.
For example, if you have an expression \((9x - 4) + (2x^2 - x + 15)\), after combining like terms, your task is to reorder them based on the power of the variables. In this case, the final result becomes \(2x^2 + 8x + 11\), because \(2x^2\) has the highest degree (2), followed by \(x\) with degree 1, and finally the constant term without any variable.
Writing polynomials in standard form helps in identifying the leading term, which is the term with the highest power, and is useful for further polynomial operations and analysis.
For example, if you have an expression \((9x - 4) + (2x^2 - x + 15)\), after combining like terms, your task is to reorder them based on the power of the variables. In this case, the final result becomes \(2x^2 + 8x + 11\), because \(2x^2\) has the highest degree (2), followed by \(x\) with degree 1, and finally the constant term without any variable.
Writing polynomials in standard form helps in identifying the leading term, which is the term with the highest power, and is useful for further polynomial operations and analysis.
Polynomial Addition
Adding polynomials involves combining terms with the same variables and exponents. This process makes it easier to simplify and solve polynomial expressions.
When performing polynomial addition, it is crucial to carefully align and match like terms. In our exercise of \((9x - 4) + (2x^2 - x + 15)\), you combine the coefficients of matching terms, ensuring you rearrange them into standard form by degree after the addition.
When performing polynomial addition, it is crucial to carefully align and match like terms. In our exercise of \((9x - 4) + (2x^2 - x + 15)\), you combine the coefficients of matching terms, ensuring you rearrange them into standard form by degree after the addition.
- Add like terms, such as \(9x\) and \(-x\), resulting in \(8x\).
- Combine constant terms, which results in \(11\) from \(-4 + 15\).
- Ensure every distinct term is included in the final expression, like \(2x^2\), even when it lacks a partner term for addition.
Other exercises in this chapter
Problem 69
Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=\sqrt{1-x}$$
View solution Problem 69
Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=|x-2|, \quad
View solution Problem 69
Use a graphing utility to graph the function. Find the domain and range of the function. $$g(x)=|2 x+3|$$
View solution Problem 69
Write the slope-intercept forms of the equations of the lines through the given point (a) parallel to the given line and (b) perpendicular to the given line. $$
View solution