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Consider the following polynomial equations:

[tex]\[
\begin{array}{l}
A = 3x^2(x-1) \\
B = -3x^3 + 4x^2 - 2x + 1
\end{array}
\][/tex]

Perform each operation and determine if the result is a polynomial.

Is the result of [tex]\[A + B\][/tex] a polynomial? [tex]\[\square\][/tex]

Is the result of [tex]\[A - B\][/tex] a polynomial? [tex]\[\square\][/tex]

Is the result of [tex]\[A \cdot B\][/tex] a polynomial? [tex]\[\square\][/tex]

Sagot :

GinnyAnswer
Let's analyze the given polynomials and their operations to determine if the results are still polynomials.

Given:
[tex]\[ A = 3x^2(x - 1) \][/tex]
[tex]\[ B = -3x^3 + 4x^2 - 2x + 1 \][/tex]

### Step-by-Step Solution:

1. Addition of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ A + B = 3x^2(x - 1) + (-3x^3 + 4x^2 - 2x + 1) \][/tex]
Combining the terms will yield a new expression in terms of [tex]\(x\)[/tex]. Since both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are polynomials, their sum [tex]\(A + B\)[/tex] is also a polynomial.

2. Subtraction of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ A - B = 3x^2(x - 1) - (-3x^3 + 4x^2 - 2x + 1) \][/tex]
Distributing and combining the terms as we did in addition will result in another polynomial. Since subtraction of polynomials results in a polynomial, [tex]\(A - B\)[/tex] is a polynomial.

3. Multiplication of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ A \cdot B = 3x^2(x - 1) \cdot (-3x^3 + 4x^2 - 2x + 1) \][/tex]
When multiplying these two polynomials, the product will contain terms that are the product of the terms in [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. Since the product of polynomials is also a polynomial, [tex]\(A \cdot B\)[/tex] is a polynomial.

### Conclusion:

Given the operations we performed:

1. Is the result of [tex]\(A + B\)[/tex] a polynomial? Yes
2. Is the result of [tex]\(A - B\)[/tex] a polynomial? Yes
3. Is the result of [tex]\(A \cdot B\)[/tex] a polynomial? Yes

So, for each question, the correct answer from the drop-down menu is "Yes".
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