Answer :
To divide the polynomial [tex]\(2x^3 + 5x^2 + 18x + 45\)[/tex] by [tex]\(2x + 5\)[/tex], we can use polynomial long division. Let's go through the process step-by-step:
1. Set up the division:
Write the dividend [tex]\(2x^3 + 5x^2 + 18x + 45\)[/tex] under the division symbol and the divisor [tex]\(2x + 5\)[/tex] outside.
2. Divide the first term:
Divide the first term of the dividend, [tex]\(2x^3\)[/tex], by the first term of the divisor, [tex]\(2x\)[/tex]:
[tex]\[
\frac{2x^3}{2x} = x^2
\][/tex]
This result, [tex]\(x^2\)[/tex], is the first term of our quotient.
3. Multiply and subtract:
Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(2x + 5\)[/tex]:
[tex]\[
x^2 \times (2x + 5) = 2x^3 + 5x^2
\][/tex]
Subtract this from the original dividend:
[tex]\[
(2x^3 + 5x^2 + 18x + 45) - (2x^3 + 5x^2) = 0x^3 + 0x^2 + 18x + 45
\][/tex]
So you are left with [tex]\(18x + 45\)[/tex].
4. Repeat the process:
Divide the next term, [tex]\(18x\)[/tex], by [tex]\(2x\)[/tex]:
[tex]\[
\frac{18x}{2x} = 9
\][/tex]
This result, [tex]\(9\)[/tex], is added to the quotient, which is now [tex]\(x^2 + 9\)[/tex].
5. Multiply and subtract again:
Multiply [tex]\(9\)[/tex] by the divisor:
[tex]\[
9 \times (2x + 5) = 18x + 45
\][/tex]
Subtract this from the remaining polynomial:
[tex]\[
(18x + 45) - (18x + 45) = 0
\][/tex]
Thus, the remainder is 0.
### Final Result
The division results in:
[tex]\[
\text{Quotient: } x^2 + 9
\][/tex]
[tex]\[
\text{Remainder: } 0
\][/tex]
This means that [tex]\(2x^3 + 5x^2 + 18x + 45\)[/tex] divides exactly by [tex]\(2x + 5\)[/tex] with a quotient of [tex]\(x^2 + 9\)[/tex] and no remainder.
1. Set up the division:
Write the dividend [tex]\(2x^3 + 5x^2 + 18x + 45\)[/tex] under the division symbol and the divisor [tex]\(2x + 5\)[/tex] outside.
2. Divide the first term:
Divide the first term of the dividend, [tex]\(2x^3\)[/tex], by the first term of the divisor, [tex]\(2x\)[/tex]:
[tex]\[
\frac{2x^3}{2x} = x^2
\][/tex]
This result, [tex]\(x^2\)[/tex], is the first term of our quotient.
3. Multiply and subtract:
Multiply [tex]\(x^2\)[/tex] by the entire divisor [tex]\(2x + 5\)[/tex]:
[tex]\[
x^2 \times (2x + 5) = 2x^3 + 5x^2
\][/tex]
Subtract this from the original dividend:
[tex]\[
(2x^3 + 5x^2 + 18x + 45) - (2x^3 + 5x^2) = 0x^3 + 0x^2 + 18x + 45
\][/tex]
So you are left with [tex]\(18x + 45\)[/tex].
4. Repeat the process:
Divide the next term, [tex]\(18x\)[/tex], by [tex]\(2x\)[/tex]:
[tex]\[
\frac{18x}{2x} = 9
\][/tex]
This result, [tex]\(9\)[/tex], is added to the quotient, which is now [tex]\(x^2 + 9\)[/tex].
5. Multiply and subtract again:
Multiply [tex]\(9\)[/tex] by the divisor:
[tex]\[
9 \times (2x + 5) = 18x + 45
\][/tex]
Subtract this from the remaining polynomial:
[tex]\[
(18x + 45) - (18x + 45) = 0
\][/tex]
Thus, the remainder is 0.
### Final Result
The division results in:
[tex]\[
\text{Quotient: } x^2 + 9
\][/tex]
[tex]\[
\text{Remainder: } 0
\][/tex]
This means that [tex]\(2x^3 + 5x^2 + 18x + 45\)[/tex] divides exactly by [tex]\(2x + 5\)[/tex] with a quotient of [tex]\(x^2 + 9\)[/tex] and no remainder.
