Select the correct answer from each drop-down menu. Consider the following polynomial equations: [tex]
\begin{array}{l}
A=3 x^2(x-1) \
B=-3 x^3+4 x^2-2 x+1
\end{array}
[/tex] Perform each operation and determine if the result is a polynomial. Is the result of [tex]$A+B$[/tex] a polynomial? Is the result of [tex]$A-B$[/tex] a polynomial? Is the result of [tex]$A \bullet B$[/tex] a polynomial?
Answer (2)
Expand A : A = 3 x 3 − 3 x 2 .
Calculate A + B = ( 3 x 3 − 3 x 2 ) + ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = x 2 − 2 x + 1 , which is a polynomial.
Calculate A − B = ( 3 x 3 − 3 x 2 ) − ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = 6 x 3 − 7 x 2 + 2 x − 1 , which is a polynomial.
Calculate A \t B = ( 3 x 3 − 3 x 2 ) ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = − 9 x 6 + 21 x 5 − 18 x 4 + 9 x 3 − 3 x 2 , which is a polynomial. Therefore, all results are polynomials.
Explanation
Understanding the Problem We are given two polynomial equations:
A = 3 x 2 ( x − 1 )
B = − 3 x 3 + 4 x 2 − 2 x + 1
We need to perform the operations A + B , A − B , and A \t B and determine if the result is a polynomial.
Expanding A First, let's expand A :
A = 3 x 2 ( x − 1 ) = 3 x 3 − 3 x 2
Calculating A+B Now, let's calculate A + B :
A + B = ( 3 x 3 − 3 x 2 ) + ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = 3 x 3 − 3 x 2 − 3 x 3 + 4 x 2 − 2 x + 1 = x 2 − 2 x + 1
The result is x 2 − 2 x + 1 , which is a polynomial.
Calculating A-B Next, let's calculate A − B :
A − B = ( 3 x 3 − 3 x 2 ) − ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = 3 x 3 − 3 x 2 + 3 x 3 − 4 x 2 + 2 x − 1 = 6 x 3 − 7 x 2 + 2 x − 1
The result is 6 x 3 − 7 x 2 + 2 x − 1 , which is a polynomial.
Calculating A*B Now, let's calculate A \t B :
A \t B = ( 3 x 3 − 3 x 2 ) ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = 3 x 3 ( − 3 x 3 + 4 x 2 − 2 x + 1 ) − 3 x 2 ( − 3 x 3 + 4 x 2 − 2 x + 1 ) = − 9 x 6 + 12 x 5 − 6 x 4 + 3 x 3 + 9 x 5 − 12 x 4 + 6 x 3 − 3 x 2 = − 9 x 6 + 21 x 5 − 18 x 4 + 9 x 3 − 3 x 2
The result is − 9 x 6 + 21 x 5 − 18 x 4 + 9 x 3 − 3 x 2 , which is a polynomial.
Determining if the results are polynomials A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Since A + B , A − B , and A \t B all result in expressions that fit this definition, they are all polynomials.
Final Answer Therefore:
The result of A + B is a polynomial. The result of A − B is a polynomial. The result of A \t B is a polynomial.
Examples
Polynomials are used to model various real-world phenomena. For example, the trajectory of a ball thrown in the air can be modeled using a quadratic polynomial. Engineers use polynomials to design curves for roads and bridges. Economists use polynomials to model cost and revenue functions. Understanding polynomial operations is crucial in many fields for making predictions and optimizing designs.
The results of the operations A + B, A - B, and A * B are all polynomials: A + B results in x^2 - 2x + 1, A - B results in 6x^3 - 7x^2 + 2x - 1, and A * B results in -9x^6 + 21x^5 - 18x^4 + 9x^3 - 3x^2.
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