(m) [tex]\frac{2}{7} y^3-\frac{1}{7} y^2+\frac{6}{7} y, \quad \frac{7}{8} y-\frac{5}{4} y^2-\frac{3}{2} y^3[/tex]
Answer (2)
Add the two polynomials by combining like terms.
Find a common denominator for each pair of coefficients.
Combine the coefficients.
The sum of the two polynomials is − 14 17 y 3 − 28 39 y 2 + 56 97 y .
Explanation
Understanding the Problem We are given two polynomials: 7 2 y 3 − 7 1 y 2 + 7 6 y and 8 7 y − 4 5 y 2 − 2 3 y 3 . The problem asks us to find the sum of these two polynomials.
Setting up the Addition Let's add the two polynomials by combining like terms. We have:
( 7 2 y 3 − 7 1 y 2 + 7 6 y ) + ( 8 7 y − 4 5 y 2 − 2 3 y 3 )
Grouping Like Terms Now, let's group the like terms together:
( 7 2 y 3 − 2 3 y 3 ) + ( − 7 1 y 2 − 4 5 y 2 ) + ( 7 6 y + 8 7 y )
Finding Common Denominators Next, we need to find a common denominator for each pair of coefficients. For the y 3 terms, the common denominator is 14. For the y 2 terms, the common denominator is 28. For the y terms, the common denominator is 56.
So we have:
( 14 4 y 3 − 14 21 y 3 ) + ( − 28 4 y 2 − 28 35 y 2 ) + ( 56 48 y + 56 49 y )
Combining Coefficients Now, we can combine the coefficients:
( 14 4 − 21 ) y 3 + ( 28 − 4 − 35 ) y 2 + ( 56 48 + 49 ) y
= ( 14 − 17 ) y 3 + ( 28 − 39 ) y 2 + ( 56 97 ) y
Final Result Therefore, the sum of the two polynomials is:
− 14 17 y 3 − 28 39 y 2 + 56 97 y
Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design roads and bridges, ensuring smooth transitions and optimal load distribution. Economists use polynomials to model cost and revenue functions, helping businesses make informed decisions about pricing and production levels. Understanding polynomial operations like addition is fundamental to these applications.
To add the two polynomials, we combine like terms by finding common denominators for the coefficients. This results in the sum of the polynomials as − 14 17 y 3 − 28 39 y 2 + 56 97 y .
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