What is the difference of the polynomials?
[tex]$\left(12 x^2-11 y^2-13 x\right)-\left(5 x^2-14 y^2-9 x\right)$[/tex]
Answer (2)
Distribute the negative sign: ( 12 x 2 − 11 y 2 − 13 x ) − ( 5 x 2 − 14 y 2 − 9 x ) = 12 x 2 − 11 y 2 − 13 x − 5 x 2 + 14 y 2 + 9 x .
Combine the x 2 terms: 12 x 2 − 5 x 2 = 7 x 2 .
Combine the y 2 terms: − 11 y 2 + 14 y 2 = 3 y 2 .
Combine the x terms: − 13 x + 9 x = − 4 x . The final answer is 7 x 2 + 3 y 2 − 4 x .
Explanation
Understanding the Problem We are given two polynomials: ( 12 x 2 − 11 y 2 − 13 x ) and ( 5 x 2 − 14 y 2 − 9 x ) . We are asked to find the difference between these two polynomials: ( 12 x 2 − 11 y 2 − 13 x ) − ( 5 x 2 − 14 y 2 − 9 x ) . We need to simplify the expression by combining like terms.
Distributing the Negative Sign To find the difference, we subtract the second polynomial from the first. This means we need to distribute the negative sign to each term in the second polynomial: ( 12 x 2 − 11 y 2 − 13 x ) − ( 5 x 2 − 14 y 2 − 9 x ) = 12 x 2 − 11 y 2 − 13 x − 5 x 2 + 14 y 2 + 9 x
Combining Like Terms Now, we combine like terms. First, let's combine the x 2 terms: 12 x 2 − 5 x 2 = ( 12 − 5 ) x 2 = 7 x 2 Next, we combine the y 2 terms: − 11 y 2 + 14 y 2 = ( − 11 + 14 ) y 2 = 3 y 2 Finally, we combine the x terms: − 13 x + 9 x = ( − 13 + 9 ) x = − 4 x
Writing the Simplified Polynomial Now we write the simplified polynomial by combining the results from the previous step: 7 x 2 + 3 y 2 − 4 x
Final Answer The difference of the polynomials is 7 x 2 + 3 y 2 − 4 x .
Examples
Polynomial subtraction is used in various applications, such as in engineering to analyze the forces acting on a structure, in computer graphics to manipulate and render images, and in economics to model cost and revenue functions. For example, if a company's revenue is modeled by the polynomial 15 x 2 + 10 x and its cost is modeled by the polynomial 8 x 2 + 3 x , then the profit can be found by subtracting the cost polynomial from the revenue polynomial: ( 15 x 2 + 10 x ) − ( 8 x 2 + 3 x ) = 7 x 2 + 7 x . This resulting polynomial represents the company's profit as a function of x .
The difference of the given polynomials is 7 x 2 + 3 y 2 − 4 x . This is achieved by distributing the negative sign and combining like terms from the two polynomials. Careful attention to each term ensures accurate results in polynomial subtraction.
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