%2F(x-5)-long-division.jpeg "(3x^2-14x-5)/(x-5) Long Division")
Long Division of Polynomials: (3x^2 - 14x - 5) / (x - 5)
Long division with polynomials is a process used to divide a polynomial by another polynomial of lesser or equal degree. It's similar to the long division you learned in elementary school, but with variables and exponents. Let's walk through the steps for dividing (3x^2 - 14x - 5) by (x - 5).
Steps:
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Set up the division:
Write the problem like a regular long division:
__________ x - 5 | 3x^2 - 14x - 5 -
Divide the leading terms:
- Focus on the leading term of the divisor (x) and the leading term of the dividend (3x^2).
- Ask yourself: "What do I multiply 'x' by to get '3x^2'?"
- The answer is 3x. Write this above the line.
- Multiply the entire divisor (x - 5) by 3x and write the result below the dividend:
3x x - 5 | 3x^2 - 14x - 5 3x^2 - 15x ------- -
Subtract:
- Subtract the terms you just wrote below the dividend. Remember to change the signs of the terms being subtracted.
- In this case, 3x^2 - 3x^2 = 0 and -14x + 15x = x. Bring down the -5:
3x x - 5 | 3x^2 - 14x - 5 3x^2 - 15x ------- x - 5 -
Repeat:
- Now focus on the new leading term of the dividend (x) and the leading term of the divisor (x).
- Ask: "What do I multiply 'x' by to get 'x'?" The answer is 1.
- Write 1 above the line next to 3x.
- Multiply (x - 5) by 1 and write the result below the x - 5:
3x + 1 x - 5 | 3x^2 - 14x - 5 3x^2 - 15x ------- x - 5 x - 5 ----- -
Subtract again:
- Subtract the terms below the line. x - x = 0 and -5 + 5 = 0. We are left with a remainder of 0:
3x + 1 x - 5 | 3x^2 - 14x - 5 3x^2 - 15x ------- x - 5 x - 5 ----- 0
Solution:
Therefore, (3x^2 - 14x - 5) divided by (x - 5) is 3x + 1.
You can check your answer by multiplying the quotient (3x + 1) by the divisor (x - 5):
(3x + 1)(x - 5) = 3x^2 - 15x + x - 5 = 3x^2 - 14x - 5
This confirms that our division was correct.