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Chapter 5: Problem 23

add the polynomials. $$\left(4 x^{2}-6 x+12\right)+\left(x^{2}+3 x+1\right)$$

Short Answer

Expert verified
The sum of the two given polynomials is \(5 x^{2}-3 x+13\).

Step by step solution

01

Identify Like Terms

In both polynomials, like terms are terms that have same power of \(x\). Here, \(4 x^{2}\) and \(x^{2}\) are like terms, \(-6 x\) and \(3 x\) are like terms and \(12\) and \(1\) are constant terms and can be directly combined.
02

Combine Like Terms

Add the coefficients of like terms together. Coefficients of \(x^{2}\) are 4 and 1, which add up to \(5 x^{2}\). Coefficients of \(x\) are -6 and 3, which add up to \(-3 x\). Constants are 12 and 1, which add up to 13. So, the combined polynomial becomes \(5 x^{2}-3 x+13\).

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Key Concepts

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

Like Terms
Like terms in polynomials are terms that have the same variable raised to the same power. This is a crucial idea when working with polynomials because it dictates which terms you can add together. When terms are not like terms, they cannot be combined through addition or subtraction.

For example, in the polynomials \(4x^2 - 6x + 12\) and \(x^2 + 3x + 1\), you can identify like terms by looking at the powers of \(x\):
  • The terms \(4x^2\) and \(x^2\) both have \(x^2\); hence, they are like terms.
  • The terms \(-6x\) and \(3x\) both have \(x\); so, they are also like terms.
  • The numbers 12 and 1 are constants, and constants with no variable part are considered like terms by default.
Understanding and identifying like terms is the first step in simplifying or combining polynomials effectively.
Coefficients
Coefficients are the numerical part of terms that include variables. They play a pivotal role in polynomial equations because they determine the magnitude of each term. When combining polynomials, you'll add or subtract these coefficients to simplify the expression.

In the polynomial addition problem given, the coefficients are important:
  • For the like terms \(x^2\), the coefficients are 4 and 1. When combined, they become \(5x^2\).
  • For the like terms \(x\), the coefficients are -6 and 3. When combined, they sum up to \(-3x\).
  • For the constant terms 12 and 1, the coefficients are the numbers themselves, which combine to give 13.
By only focusing on the coefficients, you can simplify complex-looking polynomial equations into more manageable forms.
Combine Polynomials
Combining polynomials involves adding or subtracting them to simplify into a single expression. The process requires careful attention to like terms and their respective coefficients.

Here is the step-by-step approach used for the problem:
  • First, identify the like terms. This is important because only like terms can be combined. Ignoring this step will lead to incorrect results.
  • Then, add the coefficients of these like terms. Each variable and its power need to match perfectly for a successful combination.
  • Finally, simplify by writing a single polynomial that combines all the terms. In our case, the polynomials \(4x^2 - 6x + 12\) and \(x^2 + 3x + 1\) combine to become \(5x^2 - 3x + 13\).
By following these steps, you can effectively manage and solve polynomial addition problems, making them much more accessible.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I divide monomials by dividing coefficients and subtracting exponents.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5 x^{2} \cdot 4 x^{6}=9 x^{8}$$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\left(2 x^{2}-8 x+6\right)-\left(x^{2}-3 x+5\right)=x^{2}-5 x+1\) for any value of \(x .\)

Express \(\frac{7}{8}\) as a decimal.

Find the absolute value: \(|-20.3|\)
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