Answer :
To solve the polynomial division problem, let's divide [tex]\( 3x^4 + 22x^3 + 45x^2 + 60x + 50 \)[/tex] by [tex]\( x + 5 \)[/tex].
### Steps to Solve:
1. Set up the Division:
- Your dividend is [tex]\( 3x^4 + 22x^3 + 45x^2 + 60x + 50 \)[/tex].
- Your divisor is [tex]\( x + 5 \)[/tex].
2. Perform Polynomial Long Division:
- Step 1: Divide the leading term of the dividend ([tex]\(3x^4\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]) to get [tex]\(3x^3\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x + 5\)[/tex] by this result ([tex]\(3x^3\)[/tex]), which gives [tex]\(3x^4 + 15x^3\)[/tex].
- Step 3: Subtract [tex]\(3x^4 + 15x^3\)[/tex] from the dividend:
[tex]\[
(3x^4 + 22x^3 + 45x^2 + 60x + 50) - (3x^4 + 15x^3) = 7x^3 + 45x^2 + 60x + 50
\][/tex]
- Step 4: Divide the new leading term ([tex]\(7x^3\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]), which gives [tex]\(7x^2\)[/tex].
- Step 5: Multiply the divisor by [tex]\(7x^2\)[/tex] to get [tex]\(7x^3 + 35x^2\)[/tex].
- Step 6: Subtract [tex]\(7x^3 + 35x^2\)[/tex] from the previous remainder:
[tex]\[
(7x^3 + 45x^2 + 60x + 50) - (7x^3 + 35x^2) = 10x^2 + 60x + 50
\][/tex]
- Step 7: Divide [tex]\(10x^2\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(10x\)[/tex].
- Step 8: Multiply the divisor by [tex]\(10x\)[/tex] to get [tex]\(10x^2 + 50x\)[/tex].
- Step 9: Subtract [tex]\(10x^2 + 50x\)[/tex] from [tex]\(10x^2 + 60x + 50\)[/tex]:
[tex]\[
(10x^2 + 60x + 50) - (10x^2 + 50x) = 10x + 50
\][/tex]
- Step 10: Divide [tex]\(10x\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(10\)[/tex].
- Step 11: Multiply the divisor by [tex]\(10\)[/tex] to get [tex]\(10x + 50\)[/tex].
- Step 12: Subtract [tex]\(10x + 50\)[/tex] from [tex]\(10x + 50\)[/tex], giving us a remainder of 0.
### Conclusion:
The quotient from this division is [tex]\(3x^3 + 7x^2 + 10x + 10\)[/tex], with a remainder of 0. This effective division confirms that [tex]\(3x^4 + 22x^3 + 45x^2 + 60x + 50\)[/tex] is divisible by [tex]\(x + 5\)[/tex] without any remainder.
### Steps to Solve:
1. Set up the Division:
- Your dividend is [tex]\( 3x^4 + 22x^3 + 45x^2 + 60x + 50 \)[/tex].
- Your divisor is [tex]\( x + 5 \)[/tex].
2. Perform Polynomial Long Division:
- Step 1: Divide the leading term of the dividend ([tex]\(3x^4\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]) to get [tex]\(3x^3\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x + 5\)[/tex] by this result ([tex]\(3x^3\)[/tex]), which gives [tex]\(3x^4 + 15x^3\)[/tex].
- Step 3: Subtract [tex]\(3x^4 + 15x^3\)[/tex] from the dividend:
[tex]\[
(3x^4 + 22x^3 + 45x^2 + 60x + 50) - (3x^4 + 15x^3) = 7x^3 + 45x^2 + 60x + 50
\][/tex]
- Step 4: Divide the new leading term ([tex]\(7x^3\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]), which gives [tex]\(7x^2\)[/tex].
- Step 5: Multiply the divisor by [tex]\(7x^2\)[/tex] to get [tex]\(7x^3 + 35x^2\)[/tex].
- Step 6: Subtract [tex]\(7x^3 + 35x^2\)[/tex] from the previous remainder:
[tex]\[
(7x^3 + 45x^2 + 60x + 50) - (7x^3 + 35x^2) = 10x^2 + 60x + 50
\][/tex]
- Step 7: Divide [tex]\(10x^2\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(10x\)[/tex].
- Step 8: Multiply the divisor by [tex]\(10x\)[/tex] to get [tex]\(10x^2 + 50x\)[/tex].
- Step 9: Subtract [tex]\(10x^2 + 50x\)[/tex] from [tex]\(10x^2 + 60x + 50\)[/tex]:
[tex]\[
(10x^2 + 60x + 50) - (10x^2 + 50x) = 10x + 50
\][/tex]
- Step 10: Divide [tex]\(10x\)[/tex] by [tex]\(x\)[/tex], which results in [tex]\(10\)[/tex].
- Step 11: Multiply the divisor by [tex]\(10\)[/tex] to get [tex]\(10x + 50\)[/tex].
- Step 12: Subtract [tex]\(10x + 50\)[/tex] from [tex]\(10x + 50\)[/tex], giving us a remainder of 0.
### Conclusion:
The quotient from this division is [tex]\(3x^3 + 7x^2 + 10x + 10\)[/tex], with a remainder of 0. This effective division confirms that [tex]\(3x^4 + 22x^3 + 45x^2 + 60x + 50\)[/tex] is divisible by [tex]\(x + 5\)[/tex] without any remainder.
