Add the following polynomial expressions:
$\left(g^2-4 g^4+5 g+9\right)+\left(-3 g^3+3 g^2-6\right)$
1. Rewrite terms that are subtracted as addition of the opposite.
$g^2+\left(-4 g^4\right)+5 g+9+\left(-3 g^3\right)+3 g^2+(-6)$
2. Group like terms.
3. Combine like terms.
4. Write the resulting polynomial in standard form.
Complete the steps to find the sum. What is the sum?
A. $-7 g^4+4 g^3-3 g^2+5 g-3$
B. $-4 g^4-3 g^3+4 g^2+5 g+3$
C. $-4 g^4+4 g^2+14 g-6$
D. $-3 g^4+14 g-6$
Answer (2)
To find the sum of the given polynomials, we combined like terms and arrived at the polynomial − 4 g 4 − 3 g 3 + 4 g 2 + 5 g + 3 . The correct answer matches option B. This represents the standard form of the resulting polynomial after addition.
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Remove the parentheses from the expression: g 2 − 4 g 4 + 5 g + 9 − 3 g 3 + 3 g 2 − 6 .
Group like terms: − 4 g 4 − 3 g 3 + ( g 2 + 3 g 2 ) + 5 g + ( 9 − 6 ) .
Combine like terms: − 4 g 4 − 3 g 3 + 4 g 2 + 5 g + 3 .
The sum of the polynomials is − 4 g 4 − 3 g 3 + 4 g 2 + 5 g + 3 .
Explanation
Understanding the Problem We are asked to find the sum of two polynomials: ( g 2 − 4 g 4 + 5 g + 9 ) + ( − 3 g 3 + 3 g 2 − 6 ) . The steps to add the polynomials are given, and we need to complete the steps to find the sum and choose the correct answer from the given options.
Removing Parentheses First, rewrite the expression by removing the parentheses: g 2 − 4 g 4 + 5 g + 9 − 3 g 3 + 3 g 2 − 6 .
Grouping Like Terms Next, group like terms: − 4 g 4 − 3 g 3 + ( g 2 + 3 g 2 ) + 5 g + ( 9 − 6 ) .
Combining Like Terms Now, combine like terms: − 4 g 4 − 3 g 3 + 4 g 2 + 5 g + 3 .
Final Answer Finally, write the resulting polynomial in standard form: − 4 g 4 − 3 g 3 + 4 g 2 + 5 g + 3 . Comparing this result to the given options, we see that the correct answer is − 4 g 4 − 3 g 3 + 4 g 2 + 5 g + 3 .
Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves and surfaces in computer graphics, to describe the motion of objects in physics, and to analyze data in statistics. Adding and subtracting polynomials is a fundamental skill that is used in many of these applications.
