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Chapter 5: Problem 23

Divide as indicated. Check each answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. $$\frac{x^{3}+4 x-3}{x-2}$$

Short Answer

Expert verified
The result of the division \(\frac{x^{3}+4 x-3}{x-2}\) is \(x^{2}+2x+4\) with a remainder of 5. Confirmation of this is obtained by the equation \((x-2)(x^{2}+2x+4)+5=x^{3}+4x-3\).

Step by step solution

01

Apply Polynomial Division

To divide \(x^{3}+4 x-3\) by \(x-2\), you will use the polynomial division method. Align terms of the same degree in vertical columns and perform the operation as you would perform a normal division. The result is \(x^{2}+2x+4\) and the remainder is 5.
02

Check the Result

To check the result, multiply the divisor with the quotient and then add the remainder. This operation should give you the dividend. So, check, \((x-2)(x^{2}+2x+4)+5\). After performing this operation, the result will be \(x^{3}+4x-3\).

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Most popular questions from this chapter

In Exercises \(79-82,\) simplify each expression. $$\left(\frac{18 x^{2} y^{4}}{9 x y^{2}}\right)-\left(\frac{15 x^{5} y^{6}}{5 x^{4} y^{4}}\right)$$

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In Exercises \(53-78,\) divide the polynomial by the monomial. Check each answer by showing that the product of the divisor and the quotient is the dividend. $$\frac{8 x^{6} y^{3}-12 x^{8} y^{2}-4 x^{14} y^{6}}{-4 x^{6} y^{2}}$$

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