To divide the polynomial [tex]\(10x^3 + 19x^2 + 15x - 11\)[/tex] by [tex]\(2x^2 + 5x + 6\)[/tex] using long division, we follow these steps:
1. Set up the division: Write the dividend [tex]\(10x^3 + 19x^2 + 15x - 11\)[/tex] and the divisor [tex]\(2x^2 + 5x + 6\)[/tex].
2. Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{10x^3}{2x^2} = 5x
\][/tex]
This gives us the first term of the quotient, [tex]\(5x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(2x^2 + 5x + 6\)[/tex] by [tex]\(5x\)[/tex] and subtract it from the original dividend:
[tex]\[
(10x^3 + 19x^2 + 15x - 11) - (5x \cdot (2x^2 + 5x + 6)) = (10x^3 + 19x^2 + 15x - 11) - (10x^3 + 25x^2 + 30x)
\][/tex]
[tex]\[
= (10x^3 + 19x^2 + 15x - 11) - 10x^3 - 25x^2 - 30x = -6x^2 - 15x - 11
\][/tex]
4. Repeat the process: Now we repeat the division with the new polynomial [tex]\(-6x^2 - 15x - 11\)[/tex]:
[tex]\[
\frac{-6x^2}{2x^2} = -3
\][/tex]
This gives us the next term of the quotient, [tex]\(-3\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor by [tex]\(-3\)[/tex] and subtract from the result of the previous subtraction:
[tex]\[
(-6x^2 - 15x - 11) - (-3 \cdot (2x^2 + 5x + 6)) = (-6x^2 - 15x - 11) - (-6x^2 - 15x - 18)
\][/tex]
[tex]\[
= (-6x^2 - 15x - 11) - -6x^2 - -15x - -18 = ( - 11 +18 ) = 7
\][/tex]
6. Write the result: Now we have:
[tex]\[
Quotient: 5x - 3
\][/tex]
[tex]\[
Remainder: 7
\][/tex]
So, the division of [tex]\(10x^3 + 19x^2 + 15x - 11\)[/tex] by [tex]\(2x^2 + 5x + 6\)[/tex] yields a quotient of [tex]\( 5x - 3 \)[/tex] and a remainder of [tex]\( 7 \)[/tex].
Final answer:
[tex]\[
\boxed{5x - 3 + \frac{7}{2x^2 + 5x + 6}}
\][/tex]
