wo examples of polynomial expressions are $A(x) = 4x^4 - 3x^3 + 2x - 7$ (which is of degree 4) and $B(x) = 2x^3 + 6x^2 + 1$ (which is of degree 3). To find the sum of these two polynomials, we combine like terms. The resulting polynomial is $$C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1) = 4x^4 - x^3 + 6x^2 + 2x - 6.$$ The highest degree term in this sum is $4x^4$, which gives the polynomial $C(x)$ a degree of 4.

Question

wo examples of polynomial expressions are $A(x) = 4x^4 - 3x^3 + 2x - 7$ (which is of degree 4) and $B(x) = 2x^3 + 6x^2 + 1$ (which is of degree 3). To find the sum of these two polynomials, we combine like terms. The resulting polynomial is $$C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1) = 4x^4 - x^3 + 6x^2 + 2x - 6.$$ The highest degree term in this sum is $4x^4$, which gives the polynomial $C(x)$ a degree of 4.
Next, we can compute the difference, product, and quotient of these two polynomials. The difference is $D(x) = A(x) - B(x) = 4x^4 + ( -3x^3 - 2x^3) + 6x^2 + (2x) + (-7 - 1) = 4x^4 - 5x^3 + 6x^2 + 2x - 8.$ The product is $E(x) = A(x) \cdot B(x)$, which gives us a polynomial with a degree of 7. For the quotient, since $A(x)$ has a higher degree, we perform polynomial long division: $Q(x) = \frac{A(x)}{B(x)}$. The result will be a polynomial with a degree of 1, as the leading term of $A(x)$ divided by the leading term of $B(x)$ results in an $x$ term. The exact expressions for $E(x)$ and $Q(x)$ can be derived more precisely through multiplication and division.

rewrite this in a simple way using first person to explain how you completed the expressions.

GPT-4o mini · Answer

I started with two polynomial expressions: $A(x) = 4x^4 - 3x^3 + 2x - 7$ and $B(x) = 2x^3 + 6x^2 + 1$.

To find the sum of these two polynomials, I combined like terms:

\[ C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1). \]

This simplified to:

\[ C(x) = 4x^4 - x^3 + 6x^2 + 2x - 6. \]

The highest degree term in $C(x)$ is $4x^4$, which means the degree of this polynomial is 4.

Next, I worked on finding the difference between the two polynomials. For this, I did:

\[ D(x) = A(x) - B(x) = 4x^4 + (-3x^3 - 2x^3) + 6x^2 + (2x) + (-7 - 1). \]

This resulted in:

\[ D(x) = 4x^4 - 5x^3 + 6x^2 + 2x - 8. \]

Then, I moved on to the product of the two polynomials. I found that:

\[ E(x) = A(x) \cdot B(x) \]

would give me a polynomial with a degree of 7, since the degree of $A(x)$ is 4 and the degree of $B(x)$ is 3, and I would add those degrees together.

Finally, I looked at the quotient of the two polynomials, $Q(x) = \frac{A(x)}{B(x)}$. Since the degree of $A(x)$ is higher than that of $B(x)$, I used polynomial long division. I found that the result would be of degree 1, as the leading term of $A(x)$ divided by the leading term of $B(x)$ gives me an $x$ term. I can derive the exact expressions for $E(x)$ and $Q(x)$ through detailed multiplication and division, but for now, I understand their degrees and how they relate to each other.