To solve for the quotient [tex]\( q(x) \)[/tex] and the remainder [tex]\( r(x) \)[/tex] when dividing [tex]\( a(x) = 4x^6 + 5x^5 - 6x^2 + 2x + 1 \)[/tex] by [tex]\( b(x) = x^3 \)[/tex], we can use polynomial division. Here are the detailed steps to find [tex]\( q(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
1. Division of Leading Terms:
- The leading term of [tex]\( a(x) \)[/tex] is [tex]\( 4x^6 \)[/tex].
- The leading term of [tex]\( b(x) \)[/tex] is [tex]\( x^3 \)[/tex].
- Divide [tex]\( 4x^6 \)[/tex] by [tex]\( x^3 \)[/tex], which gives [tex]\( 4x^3 \)[/tex].
2. Multiply and Subtract:
- Multiply [tex]\( b(x) = x^3 \)[/tex] by [tex]\( 4x^3 \)[/tex], giving [tex]\( 4x^6 \)[/tex].
- Subtract [tex]\( 4x^6 \)[/tex] from [tex]\( a(x) \)[/tex]:
[tex]\[
(4x^6 + 5x^5 - 6x^2 + 2x + 1) - 4x^6 = 5x^5 - 6x^2 + 2x + 1
\][/tex]
3. Next Leading Term:
- The leading term of the resultant polynomial is [tex]\( 5x^5 \)[/tex].
- Divide [tex]\( 5x^5 \)[/tex] by [tex]\( x^3 \)[/tex], which gives [tex]\( 5x^2 \)[/tex].
4. Multiply and Subtract:
- Multiply [tex]\( b(x) = x^3 \)[/tex] by [tex]\( 5x^2 \)[/tex], giving [tex]\( 5x^5 \)[/tex].
- Subtract [tex]\( 5x^5 \)[/tex] from the current polynomial:
[tex]\[
(5x^5 - 6x^2 + 2x + 1) - 5x^5 = - 6x^2 + 2x + 1
\][/tex]
5. Conclusion:
- Since the degree of the remaining polynomial [tex]\( -6x^2 + 2x + 1 \)[/tex] is less than the degree of [tex]\( b(x) = x^3 \)[/tex], we stop here.
Thus, the quotient polynomial [tex]\( q(x) \)[/tex] is:
[tex]\[
q(x) = 4x^3 + 5x^2
\][/tex]
And the remainder polynomial [tex]\( r(x) \)[/tex] is:
[tex]\[
r(x) = -6x^2 + 2x + 1
\][/tex]
So, the final answers are:
[tex]\[
q(x) = 4x^3 + 5x^2
\][/tex]
[tex]\[
r(x) = -6x^2 + 2x + 1
\][/tex]
