Certainly! Let's perform the polynomial division of \( (4x^2 + 19x + 19) \div (x + 3) \).
1. Set up the division:
The first polynomial \( 4x^2 + 19x + 19 \) is the numerator (dividend), and \( x + 3 \) is the denominator (divisor). We need to divide the numerator by the divisor.
2. Divide the leading term of the numerator by the leading term of the divisor:
- The leading term of the numerator is \( 4x^2 \).
- The leading term of the divisor is \( x \).
- Divide \( 4x^2 \) by \( x \): [tex]\[ \frac{4x^2}{x} = 4x \][/tex]
- So, the first term of the quotient is \( 4x \).
3. Multiply the entire divisor by this quotient term:
- Multiply \( x + 3 \) by \( 4x \): [tex]\[ (x + 3) \cdot 4x = 4x^2 + 12x \][/tex]
4. Subtract the result from the original numerator:
[tex]\[
(4x^2 + 19x + 19) - (4x^2 + 12x) = 4x^2 + 19x + 19 - 4x^2 - 12x = 7x + 19
\][/tex]
5. Repeat the division process for the new polynomial:
- Divide \( 7x \) by \( x \): [tex]\[ \frac{7x}{x} = 7 \][/tex]
- So, the next term of the quotient is \( 7 \).
6. Multiply the entire divisor by this new quotient term:
- Multiply \( x + 3 \) by \( 7 \): [tex]\[ (x + 3) \cdot 7 = 7x + 21 \][/tex]
7. Subtract the result from the new polynomial:
[tex]\[
(7x + 19) - (7x + 21) = 7x + 19 - 7x - 21 = -2
\][/tex]
8. Combine the results:
- The quotient is the combination of both terms we found: \( 4x + 7 \).
- The remainder is the result we obtained from the last subtraction: \( -2 \).
So, the final quotient and remainder for the division of \( 4x^2 + 19x + 19 \) by \( x + 3 \) are:
[tex]\[
\text{Quotient: } 4x + 7
\][/tex]
[tex]\[
\text{Remainder: } -2
\][/tex]
