Divide using long division. Check your answer.
(5x^2-13x+1)/(x-3)
To divide using long division, we will first write the expression in descending order of powers of x:
5x^2 - 13x + 1 / x - 3
Next, we will divide the first term of the numerator (5x^2) by the first term of the denominator (x).
5x^2 / x = 5x.
We then multiply the entire denominator by this quotient (x - 3) and subtract it from the numerator.
5x * (x - 3) = 5x^2 - 15x
(5x^2 - 13x + 1) - (5x^2 - 15x) = (5x^2 - 5x^2) + (-13x + 15x) + 1
= 0 + 2x + 1
Now we repeat the process with the new numerator (2x + 1) and the same denominator (x - 3).
2x / x = 2.
2 * (x - 3) = 2x - 6.
(2x + 1) - (2x - 6) = 2x - 2x + 1 + 6
= 7.
We don't have any more terms in the numerator, so the division is complete.
Therefore, the quotient is 5x + 2 with a remainder of 7.
Checking our answer, if we multiply the quotient (5x + 2) by the denominator (x - 3) and add the remainder (7), we should get the original numerator (5x^2 - 13x + 1).
(5x + 2) * (x - 3) + 7
= 5x^2 - 15x + 2x - 6 + 7
= 5x^2 - 13x + 1
Therefore, our answer is correct.
To divide using long division, we will follow these steps:
Step 1: Arrange the dividend and divisor in long division format:
______
x - 3 | 5x² - 13x + 1
Step 2: Divide the first term of the dividend by the first term of the divisor. In this case, divide 5x² by x, which gives us 5x.
5x
__________
x - 3 | 5x² - 13x + 1
Step 3: Multiply the quotient (5x) by the divisor (x - 3) and subtract it from the dividend:
5x
__________
x - 3 | 5x² - 13x + 1
-(5x² - 15x)
___________
2x - 1
Step 4: Bring down the next term of the dividend, which is -1:
5x + 2
__________
x - 3 | 5x² - 13x + 1
-(5x² - 15x)
___________
2x - 1
-(2x - 6)
___________
5
Step 5: Since the next term is 5, which is a constant, there is nothing else to bring down. Therefore, the long division is complete.
Step 6: Write the final quotient as the answer. In this case, the quotient is 5x + 2 and the remainder is 5.
Therefore, (5x^2 - 13x + 1) / (x - 3) = 5x + 2 + (5 / (x - 3))