Problem 156 Find the quotient and remainder ... [FREE SOLUTION]

Chapter 8: Problem 156

Find the quotient and remainder when \(3 x^{7}-x^{6}+31 x^{4}+21 x+5\) is divided by \(x+2\)

Short Answer

Expert verified

The quotient is \(Q(x) = 3x^6 - 7x^5 + 15x^4 -30x^3 + 31x^2 -29x + 58\) and the remainder is \(R(x) = -114\) when \(3x^7-x^6+31x^4+21x+5\) is divided by \(x+2\).

Step by step solution

Set up the long division problem

First, set up the division problem by writing the dividend, \(3x^7-x^6+31x^4+21x+5\), and the divisor, \(x + 2\), like you would in a standard long division problem.

Divide the leading term of the dividend by the leading term of the divisor

In our case, the leading terms are \(3x^7\) (dividend) and \(x\) (divisor). Divide \(3x^7\) by \(x\) to get \(3x^6\). Write \(3x^6\) as the first term of the quotient.

Multiply the divisor by the term obtained in the previous step

Now, multiply the divisor, \(x + 2\), by the term we got in step 2, \(3x^6\). This will give us: \(3x^6(x + 2) = 3x^7+ 6x^6\).

Subtract the result from the dividend

Subtract the product from step 3, \(3x^7+6x^6\), from the dividend, \(3x^7-x^6+31x^4+21x+5\). This results in: \((-7x^6+31x^4+21x+5) \).

Repeat steps 2-4

Repeat steps 2-4 until the degree of the remaining polynomial is less than the degree of the divisor. Divide the leading term of the new dividend (remainder after subtraction) by the leading term of the divisor: \(\frac{-7x^6}{x} = -7x^5\). Multiply the divisor by the term obtained in the previous step and then subtract this from the dividend. After the necessary repetitions, we arrive at a final quotient and remainder.

Write down the quotient and remainder

After following the steps, we obtain the quotient: \( Q(x) = 3x^6 - 7x^5 + 15x^4 -30x^3 + 31x^2 -29x + 58 \) And the remainder: \( R(x) = -114 \) Thus, when \(3x^7-x^6+31x^4+21x+5\) is divided by \(x+2\), the quotient is \(3x^6 - 7x^5 + 15x^4 -30x^3 + 31x^2 -29x + 58\) and the remainder is \(-114\).

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91影视

Find the quotient and remainder when \(3 x^{7}-x^{6}+31 x^{4}+21 x+5\) is divided by \(x+2\)

Short Answer

Step by step solution

Set up the long division problem

Divide the leading term of the dividend by the leading term of the divisor

Multiply the divisor by the term obtained in the previous step

Subtract the result from the dividend

Repeat steps 2-4

Write down the quotient and remainder

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