%C3%B7(x2%2B2x%2B1).jpeg "(x3+x2+3x−4)÷(x2+2x+1)")
Dividing Polynomials: (x³ + x² + 3x - 4) ÷ (x² + 2x + 1)
This article will guide you through the process of dividing the polynomial (x³ + x² + 3x - 4) by (x² + 2x + 1) using long division.
Step 1: Set up the Long Division
First, write the dividend (x³ + x² + 3x - 4) inside the division symbol and the divisor (x² + 2x + 1) outside.
________
x² + 2x + 1 | x³ + x² + 3x - 4
Step 2: Divide the Leading Terms
Focus on the leading terms of both the divisor and the dividend. Divide the leading term of the dividend (x³) by the leading term of the divisor (x²). This gives us x.
x
x² + 2x + 1 | x³ + x² + 3x - 4
Step 3: Multiply and Subtract
Multiply the quotient (x) by the divisor (x² + 2x + 1) and write the result below the dividend.
x
x² + 2x + 1 | x³ + x² + 3x - 4
x³ + 2x² + x
Subtract the result from the dividend.
x
x² + 2x + 1 | x³ + x² + 3x - 4
x³ + 2x² + x
---------
-x² + 2x - 4
Step 4: Repeat Steps 2 and 3
Bring down the next term (-4) from the dividend. Focus on the leading term of the new dividend (-x²) and divide it by the leading term of the divisor (x²). This gives us -1.
x - 1
x² + 2x + 1 | x³ + x² + 3x - 4
x³ + 2x² + x
---------
-x² + 2x - 4
-x² - 2x - 1
Multiply -1 by the divisor and subtract the result.
x - 1
x² + 2x + 1 | x³ + x² + 3x - 4
x³ + 2x² + x
---------
-x² + 2x - 4
-x² - 2x - 1
---------
4x - 3
Step 5: Interpret the Result
Since the degree of the new dividend (4x - 3) is less than the degree of the divisor (x² + 2x + 1), we stop here.
Therefore, the result of dividing (x³ + x² + 3x - 4) by (x² + 2x + 1) is:
x - 1 + (4x - 3) / (x² + 2x + 1)
This means that the quotient is x - 1 and the remainder is 4x - 3.