%2F(x-5).jpeg "(3x^3-17x^2+15x-25)/(x-5)")
Dividing Polynomials: (3x^3 - 17x^2 + 15x - 25) / (x - 5)
This article explores the division of the polynomial (3x^3 - 17x^2 + 15x - 25) by (x - 5) using polynomial long division.
Polynomial Long Division
Polynomial long division is a method used to divide polynomials. It is similar to the long division method used for dividing numbers.
Step 1: Set up the division.
Write the dividend (3x^3 - 17x^2 + 15x - 25) inside the division symbol and the divisor (x - 5) outside the division symbol.
_________
x - 5 | 3x^3 - 17x^2 + 15x - 25
Step 2: Divide the leading terms.
Divide the leading term of the dividend (3x^3) by the leading term of the divisor (x). This gives us 3x^2.
3x^2
x - 5 | 3x^3 - 17x^2 + 15x - 25
Step 3: Multiply the divisor by the quotient.
Multiply the divisor (x - 5) by the quotient (3x^2) to get 3x^3 - 15x^2.
3x^2
x - 5 | 3x^3 - 17x^2 + 15x - 25
-(3x^3 - 15x^2)
Step 4: Subtract.
Subtract the result from the dividend.
3x^2
x - 5 | 3x^3 - 17x^2 + 15x - 25
-(3x^3 - 15x^2)
-2x^2 + 15x
Step 5: Bring down the next term.
Bring down the next term of the dividend (15x).
3x^2
x - 5 | 3x^3 - 17x^2 + 15x - 25
-(3x^3 - 15x^2)
-2x^2 + 15x - 25
Step 6: Repeat steps 2-5.
Divide the new leading term (-2x^2) by the leading term of the divisor (x). This gives us -2x. Multiply the divisor by the new quotient (-2x) and subtract. Bring down the next term (-25).
3x^2 - 2x
x - 5 | 3x^3 - 17x^2 + 15x - 25
-(3x^3 - 15x^2)
-2x^2 + 15x - 25
-(-2x^2 + 10x)
5x - 25
Step 7: Repeat steps 2-5 again.
Divide the new leading term (5x) by the leading term of the divisor (x). This gives us 5. Multiply the divisor by the new quotient (5) and subtract.
3x^2 - 2x + 5
x - 5 | 3x^3 - 17x^2 + 15x - 25
-(3x^3 - 15x^2)
-2x^2 + 15x - 25
-(-2x^2 + 10x)
5x - 25
-(5x - 25)
0
Result:
The quotient is 3x^2 - 2x + 5 and the remainder is 0. Therefore:
(3x^3 - 17x^2 + 15x - 25) / (x - 5) = 3x^2 - 2x + 5