To find the sum of the polynomials [tex]\( p(x) = x^3 - 5x^2 + x + 2 \)[/tex] and [tex]\( q(x) = x^3 - 3x^2 + 2x + 1 \)[/tex], follow these steps:
1. Express the polynomials in standard form:
[tex]\[
p(x) = x^3 - 5x^2 + x + 2
\][/tex]
[tex]\[
q(x) = x^3 - 3x^2 + 2x + 1
\][/tex]
2. Align the polynomials horizontally to combine like terms:
[tex]\[
\begin{array}{rrrr}
x^3 & -5x^2 & +x & +2 \\
+ x^3 & -3x^2 & +2x & +1 \\
\end{array}
\][/tex]
3. Add the coefficients of like terms:
- For [tex]\( x^3 \)[/tex] terms: [tex]\( 1x^3 + 1x^3 = 2x^3 \)[/tex]
- For [tex]\( x^2 \)[/tex] terms: [tex]\( -5x^2 - 3x^2 = -8x^2 \)[/tex]
- For [tex]\( x \)[/tex] terms: [tex]\( 1x + 2x = 3x \)[/tex]
- For the constant terms: [tex]\( 2 + 1 = 3 \)[/tex]
4. Write the resulting polynomial:
[tex]\[
p(x) + q(x) = 2x^3 - 8x^2 + 3x + 3
\][/tex]
5. Determine the degree of the resulting polynomial:
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] with a non-zero coefficient. For [tex]\( 2x^3 - 8x^2 + 3x + 3 \)[/tex], the highest power of [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex].
So, the sum of the polynomials is [tex]\( 2x^3 - 8x^2 + 3x + 3 \)[/tex] and the degree of the sum is [tex]\( 3 \)[/tex].
