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Chapter 5: Problem 24

Add the polynomials. $$\left(-7 x^{2}+8 x+3\right)+\left(2 x^{2}+x+8\right)$$

Short Answer

Expert verified
The sum of the given polynomials is: -5\(x^{2}\) + 9\(x\) + 11.

Step by step solution

01

Add Coefficients of \(x^{2}\)

The coefficients of \(x^{2}\) in our two polynomials are -7 and 2. Thus, add -7 and 2 to get -5. So, the sum for the \(x^{2}\) terms would be -5\(x^{2}\).
02

Add Coefficients of \(x\)

The coefficients of \(x\) in our two polynomials are 8 and 1. Thus, add 8 and 1 to get 9. So, the sum for the \(x\) terms would be 9\(x\).
03

Add Constants

The constants in our two polynomials are 3 and 8. Thus, add 3 and 8 to get 11.
04

Combine All Sums

Combine the sums from steps 1-3 to get the final polynomial. So we get -5\(x^{2}\) + 9\(x\) + 11.

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Key Concepts

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

Coefficients
When dealing with polynomials, coefficients are the numerical factors that multiply the variable part in each term. In the polynomial expression \(-7x^2 + 8x + 3\), the coefficients are the numbers in front of the variables. Here, the coefficient for the \(x^2\) term is -7, while the coefficient for the \(x\) term is 8. The term 3 is a constant and doesn't have a variable, so it's often not referred to as having a coefficient.
  • Coefficients dictate the steepness or direction of a curve when graphed.
  • In addition and subtraction of polynomials, always match terms with the same degree to correctly add or subtract their coefficients.
  • For instance, when adding \(-7x^2 + 8x + 3\) and \(2x^2 + x + 8\), focus on combining coefficients from matching powers of \(x\).
Adding the coefficients means adding the numbers in front of the same degree terms. In this exercise, the coefficient for \(x^2\) terms is calculated as: add -7 and 2 to result in -5.
Combining Like Terms
Combining like terms involves merging terms that have the same variables and exponents. This process simplifies polynomial expressions by adding or subtracting these similar terms. Given the polynomials \(-7x^2 + 8x + 3\) and \(2x^2 + x + 8\), we need to look for terms with the same powers of \(x\) and combine them.
This exercise demonstrates combining each:
  • For the \(x^2\) terms: Combine -7 from \(-7x^2\) and 2 from \(2x^2\) to get \(-5x^2\).
  • For the \(x\) terms: Combine 8 from \(8x\) and 1 from \(x\) to result in \(9x\).
  • For constants, simply add them as they are already like terms.
By effectively combining like terms, you make polynomial addition straightforward and clear.
Constant Terms
Constant terms in polynomials are numbers without variables. They stand alone in the expression and are crucial in determining the polynomial's output when all variables equal zero. In our given polynomials, 3 and 8 are the constant terms.
When adding polynomials, constant terms are treated just like combining like terms, albeit without any variable. This is particularly straightforward as you only need to add the numerical values directly:
  • From our example, add 3 and 8 to result in 11. This sum is the constant term of the resulting polynomial.
  • Unlike terms with variables, constants will always combine in this way regardless of the variables' absence.
Hence, understanding constant terms aids in simplifying polynomials and finding their simplest form, providing an easier route to solve polynomial expressions and equations.

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Most popular questions from this chapter

Use a vertical format to find each product. $$\begin{array}{l}x^{2}+6 x-4 \\\x^{2}-x-2 \\\\\hline\end{array}$$

Use a vertical format to find each product. $$\begin{array}{r} 9 y^{3}-7 y^{2}+5 y \\ 3 y^{2}+5 y \\ \hline \end{array}$$

Explain how to divide a polynomial that is not a monomial by a monomial. Give an example.

Perform the indicated operations. $$(y+1)\left(y^{2}-y+1\right)-(y-1)\left(y^{2}+y+1\right)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$534.7=5.347 \times 10^{3}$$
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