Question

Find a polynomial which, when added to the polynomial 3x2+3x–1, is equivalent to the following expressions.

Answer 1

a) The polynomial $R(x) = -3\cdot x^{2}-3\cdot x$ is required to obtain the polynomial $P(x) = 1$.

b) The polynomial $R(x) = -3\cdot x^{2}-2\cdot x + 6$ is required to obtain the polynomial $P(x) = x + 5$.

In this question we must take advantage of the closure properties for the addition between two polynomials to determine all required polynomials, which are defined by this expression:

$R(x) = P(x) - Q(x)$ (1)

Where:

• $P(x)$ - Resulting polynomial.
• $Q(x)$ - Original polynomial.
• $R(x)$ - Required polynomial.

Now we proceed to determine each required polynomial:

a) $P(x) = 1$, $Q(x) = 3\cdot x^{2}+3\cdot x - 1$

$R(x) = -3\cdot x^{2}-3\cdot x$ (1)

The polynomial $R(x) = -3\cdot x^{2}-3\cdot x$ is required to obtain the polynomial $P(x) = 1$. $\blacksquare$

b) $P(x) = x + 5$, $Q(x) = 3\cdot x^{2}+3\cdot x - 1$

$R(x) = -3\cdot x^{2}-2\cdot x + 6$ (2)

The polynomial $R(x) = -3\cdot x^{2}-2\cdot x + 6$ is required to obtain the polynomial $P(x) = x + 5$. $\blacksquare$

Remark

The statement is incomplete, complete form is shown below:

Find a polynomial which, when added to the polynomial $3\cdot x^{2}+3\cdot x - 1$ is equivalent to the following expressions: (a) $1$, (b) $x + 5$.

To learn more on polynomials, we kindly invite to check this verified question: brainly.com/question/17822016