Two examples of polynomial expressions are $P(x) = 3x^3 + 2x^2 - x + 4$ (which is of degree 3) and $Q(x) = 5x^4 + 7x^2 + 2$ (which is of degree 4). When we add these polynomials together to find their sum, we combine the like terms. The resulting polynomial is $S(x) = P(x) + Q(x) = 5x^4 + 3x^3 + 9x^2 - x + 6$, and the highest degree term in this sum is $5x^4$, which gives the polynomial $S(x)$ a degree of 4.

Question

Two examples of polynomial expressions are $P(x) = 3x^3 + 2x^2 - x + 4$ (which is of degree 3) and $Q(x) = 5x^4 + 7x^2 + 2$ (which is of degree 4). When we add these polynomials together to find their sum, we combine the like terms. The resulting polynomial is $S(x) = P(x) + Q(x) = 5x^4 + 3x^3 + 9x^2 - x + 6$, and the highest degree term in this sum is $5x^4$, which gives the polynomial $S(x)$ a degree of 4.
To find the first and last terms of a polynomial, we look for the leading term, which is the term with the highest exponent, and the constant term, which is the term without any variable. In standard polynomial form, this process is straightforward. However, if the polynomial is presented in a more complex or non-standard format, additional steps might be necessary to identify the first and last terms.

rewrite this choosing two different expressions.

GPT-4o mini · Answer

Let's consider two different polynomial expressions: $A(x) = 4x^5 - 3x^3 + 2x - 7$ (which is of degree 5) and $B(x) = 2x^3 + 6x^2 + 1$ (which is of degree 3). When we add these polynomials together to find their sum, we combine the like terms. The resulting polynomial is

\[ C(x) = A(x) + B(x) = 4x^5 + ( -3x^3 + 2x^3) + 6x^2 + (2x) - 7 + 1 = 4x^5 - x^3 + 6x^2 + 2x - 6. \]

The highest degree term in this sum is $4x^5$, which gives the polynomial $C(x)$ a degree of 5.

To identify the first and last terms of the polynomial, we look for the leading term (the term with the highest exponent) and the constant term (the term without any variable). In standard polynomial form, this process is typically straightforward. However, if the polynomial is presented in a more complex or non-standard format, additional steps might be necessary to accurately identify the first and last terms.