Hart Logan
03/28/2024 · High School
Find the quotient using long division. \[ \frac{7 x^{2}+26 x+15}{x+3} \]
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Tutor-Verified Answer
Step-by-step Solution
To perform long division on the polynomial \( \frac{7x^2 + 26x + 15}{x + 3} \), we will follow the steps of polynomial long division.
1. **Divide the leading term**: Divide the leading term of the dividend \( 7x^2 \) by the leading term of the divisor \( x \):
\[
\frac{7x^2}{x} = 7x
\]
This gives us the first term of the quotient.
2. **Multiply and subtract**: Multiply \( 7x \) by the entire divisor \( x + 3 \):
\[
7x(x + 3) = 7x^2 + 21x
\]
Now, subtract this from the original polynomial:
\[
(7x^2 + 26x + 15) - (7x^2 + 21x) = (26x - 21x) + 15 = 5x + 15
\]
3. **Repeat the process**: Now, we need to divide the new leading term \( 5x \) by the leading term of the divisor \( x \):
\[
\frac{5x}{x} = 5
\]
This gives us the next term of the quotient.
4. **Multiply and subtract again**: Multiply \( 5 \) by the entire divisor \( x + 3 \):
\[
5(x + 3) = 5x + 15
\]
Now, subtract this from the current polynomial:
\[
(5x + 15) - (5x + 15) = 0
\]
Since the remainder is \( 0 \), we have completed the division.
The final result of the long division is:
\[
\frac{7x^2 + 26x + 15}{x + 3} = 7x + 5
\]
Thus, the quotient is:
\[
\boxed{7x + 5}
\]
Quick Answer
The quotient is \( 7x + 5 \).
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