fill in the missing values below to find the quotient when $x^4 + 6x^3 + 5x^2 - 4x - 20$ is divided by $x +…

fill in the missing values below to find the quotient when $x^4 + 6x^3 + 5x^2 - 4x - 20$ is divided by $x + 5$.

fill in the missing values below to find the quotient when $x^4 + 6x^3 + 5x^2 - 4x - 20$ is divided by $x + 5$.

Answer

Explanation:

We can use polynomial long division or synthetic division to solve this. Let's use polynomial long division.

Step1: Divide the leading term

Divide the leading term of the dividend (x^4) by the leading term of the divisor (x), we get (x^3).

Step2: Multiply and subtract

Multiply the divisor (x + 5) by (x^3) to get (x^4+5x^3). Subtract this from the dividend: ((x^4 + 6x^3+5x^2-4x - 20)-(x^4+5x^3)=x^3+5x^2).

Step3: Repeat the process

Now, divide the leading term (x^3) by (x) to get (x^2). Multiply the divisor by (x^2) to get (x^3 + 5x^2). Subtract: ((x^3+5x^2)-(x^3 + 5x^2)=0), and we bring down the next term (-4x). Now we have (-4x).

Step4: Continue dividing

Divide (-4x) by (x) to get (-4). Multiply the divisor by (-4) to get (-4x-20). Subtract: ((-4x - 20)-(-4x - 20)=0).

So the quotient is (x^3+x^2 - 4). Let's fill in the boxes (assuming the long - division layout):

  • The first row (quotient terms): (x^3), (x^2), (0x) (or just leave it blank as the coefficient is 0), (-4)
  • The second row (divisor and dividend terms): The divisor is (x), and the dividend coefficients are (1) (for (x^4)), (6) (for (x^3)), (5) (for (x^2)), (-4) (for (x)), (-20) (constant term)
  • The third row (multiplication results): When we multiply (x) by (x^3) we get (x^4), by (x^2) we get (x^3), by (0) we get (0), by (-4) we get (-4x); and when we multiply (5) by (x^3) we get (5x^3), by (x^2) we get (5x^2), by (0) we get (0), by (-4) we get (-20) (but in long division, we usually do the subtraction step - by - step)

But if we consider the synthetic division (since the divisor is of the form (x + a) where (a=- 5)):

The coefficients of the dividend are (1,6,5,-4,-20) and (a=-5)

  • Bring down the (1)
  • Multiply (1\times(-5)=-5), add to (6): (6+( - 5)=1)
  • Multiply (1\times(-5)=-5), add to (5): (5+( - 5)=0)
  • Multiply (0\times(-5)=0), add to (-4): (-4 + 0=-4)
  • Multiply (-4\times(-5)=20), add to (-20): (-20 + 20=0)

The synthetic division results (quotient coefficients) are (1,1,0,-4), so the quotient is (x^3+x^2-4)

Answer:

The quotient is (x^{3}+x^{2}-4)