Randy Divides $\left(2x^4 - 3x^3 - 3x^2 + 7x - 3\right$\] By $\left(x^2 - 2x + 1\right$\] As Shown Below. What Error Does Randy Make?$\[ \begin{array}{r} \frac {2x^2 + X + 3}{2} - 2x + 1 \longdiv 2x^4 - 3x^3 - 3x^2 + 7x - 3

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Understanding Polynomial Division

Polynomial division is a process of dividing a polynomial by another polynomial. It is an essential concept in algebra and is used to simplify complex expressions. In this article, we will discuss the error made by Randy in dividing a polynomial.

Randy's Division

Randy divides (2x4−3x3−3x2+7x−3)\left(2x^4 - 3x^3 - 3x^2 + 7x - 3\right) by (x2−2x+1)\left(x^2 - 2x + 1\right) as shown below.

2x2+x+32−2x+1\longdiv2x4−3x3−3x2+7x−3{ \begin{array}{r} \frac {2x^2 + x + 3}{2} - 2x + 1 \longdiv 2x^4 - 3x^3 - 3x^2 + 7x - 3 \end{array} }

Error in Division

The error in Randy's division is that he has not performed the division correctly. To perform polynomial division, we need to follow the steps of division, which include dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the degree of the remainder is less than the degree of the divisor.

Correct Division

Let's perform the correct division of (2x4−3x3−3x2+7x−3)\left(2x^4 - 3x^3 - 3x^2 + 7x - 3\right) by (x2−2x+1)\left(x^2 - 2x + 1\right).

Step 1: Divide the Highest Degree Term

The highest degree term of the dividend is 2x42x^4 and the highest degree term of the divisor is x2x^2. Therefore, we divide 2x42x^4 by x2x^2 to get 2x22x^2.

Step 2: Multiply the Divisor by the Result

We multiply the entire divisor x2−2x+1x^2 - 2x + 1 by 2x22x^2 to get 2x4−4x3+2x22x^4 - 4x^3 + 2x^2.

Step 3: Subtract the Product from the Dividend

We subtract the product 2x4−4x3+2x22x^4 - 4x^3 + 2x^2 from the dividend 2x4−3x3−3x2+7x−32x^4 - 3x^3 - 3x^2 + 7x - 3 to get x3−5x2+7x−3x^3 - 5x^2 + 7x - 3.

Step 4: Repeat the Process

We repeat the process by dividing the highest degree term of the new dividend x3−5x2+7x−3x^3 - 5x^2 + 7x - 3 by the highest degree term of the divisor x2−2x+1x^2 - 2x + 1 to get x−3x - 3.

Step 5: Multiply the Divisor by the Result

We multiply the entire divisor x2−2x+1x^2 - 2x + 1 by x−3x - 3 to get x3−5x2+7x−3x^3 - 5x^2 + 7x - 3.

Step 6: Subtract the Product from the Dividend

We subtract the product x3−5x2+7x−3x^3 - 5x^2 + 7x - 3 from the new dividend x3−5x2+7x−3x^3 - 5x^2 + 7x - 3 to get 00.

Conclusion

In conclusion, Randy's error in dividing the polynomial was that he did not perform the division correctly. He did not follow the steps of division, which include dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the degree of the remainder is less than the degree of the divisor.

Final Answer

The final answer is that Randy's error was not performing the division correctly. The correct division of (2x4−3x3−3x2+7x−3)\left(2x^4 - 3x^3 - 3x^2 + 7x - 3\right) by (x2−2x+1)\left(x^2 - 2x + 1\right) is 2x2+x+32x^2 + x + 3 with a remainder of 00.

Key Takeaways

  • Polynomial division is a process of dividing a polynomial by another polynomial.
  • To perform polynomial division, we need to follow the steps of division, which include dividing the highest degree term of the dividend by the highest degree term of the divisor, multiplying the entire divisor by the result, subtracting the product from the dividend, and repeating the process until the degree of the remainder is less than the degree of the divisor.
  • Randy's error in dividing the polynomial was that he did not perform the division correctly.
  • The correct division of (2x4−3x3−3x2+7x−3)\left(2x^4 - 3x^3 - 3x^2 + 7x - 3\right) by (x2−2x+1)\left(x^2 - 2x + 1\right) is 2x2+x+32x^2 + x + 3 with a remainder of 00.