-divided-by-(x-5).jpeg "(3x^2-14x-5) Divided By (x-5)")
Dividing Polynomials: (3x^2 - 14x - 5) ÷ (x - 5)
This article will guide you through the process of dividing the polynomial (3x^2 - 14x - 5) by the binomial (x - 5). We will use the long division method to accomplish this.
Understanding Polynomial Long Division
Long division with polynomials is very similar to long division with numbers. The main goal is to find the quotient (the result of the division) and the remainder (any leftover terms).
Steps to Divide
-
Set up the division:
- Write the dividend (3x^2 - 14x - 5) inside the division symbol.
- Write the divisor (x - 5) outside the division symbol.
___________ x - 5 | 3x^2 - 14x - 5 -
Focus on the leading terms:
- Divide the leading term of the dividend (3x^2) by the leading term of the divisor (x). This gives us 3x.
- Write 3x above the division symbol.
3x _______ x - 5 | 3x^2 - 14x - 5 -
Multiply the quotient by the divisor:
- Multiply the quotient (3x) by the divisor (x - 5). This gives us 3x^2 - 15x.
3x _______ x - 5 | 3x^2 - 14x - 5 3x^2 - 15x -
Subtract:
- Subtract (3x^2 - 15x) from the dividend. Remember to distribute the negative sign.
3x _______ x - 5 | 3x^2 - 14x - 5 3x^2 - 15x -------- x - 5 -
Bring down the next term:
- Bring down the next term from the dividend (-5).
3x _______ x - 5 | 3x^2 - 14x - 5 3x^2 - 15x -------- x - 5 -
Repeat steps 2-5:
- Divide the new leading term (x) by the leading term of the divisor (x). This gives us 1.
- Write 1 above the division symbol.
- Multiply 1 by the divisor (x - 5) to get x - 5.
- Subtract (x - 5) from the previous result.
3x + 1 ______ x - 5 | 3x^2 - 14x - 5 3x^2 - 15x -------- x - 5 x - 5 ---- 0
Result
Since we have a remainder of 0, the division is complete. The quotient is 3x + 1. This means:
(3x^2 - 14x - 5) ÷ (x - 5) = 3x + 1