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Chapter 5: Problem 86

Use a vertical format to find each product. $$\begin{array}{r}7 x^{3}-5 x^{2}+6 x \\\3 x^{2}-4 x \\\\\hline\end{array}$$

Short Answer

Expert verified
The product is 21\(x^5\) -43\(x^4\) + 38\(x^3\) - 24\(x^2\)

Step by step solution

01

Multiply the first term of the first polynomial by every term in the second polynomial

Starting with the first term 7\(x^3\), multiply by each term in the second polynomial: \n7\(x^3\) * 3\(x^2\) = 21\(x^5\)\n7\(x^3\) * -4\(x\) = -28\(x^4\)
02

Multiply the second term of the first polynomial by every term in the second polynomial

Now do the same with the second term -5\((x^2)\) from the first polynomial: \n-5\((x^2)\) * 3\((x^2)\) = -15\((x^4)\) \n-5\((x^2)\) * -4\(x\) = 20\((x^3)\)
03

Multiply the third term of the first polynomial by every term in the second polynomial

Finally, with the third term 6\(x\) from the first polynomial: \n6\(x\) * 3\((x^2)\) = 18\((x^3)\) \n6\(x\) * -4\(x\) = -24\(x^2)\)
04

Combine like terms

Combine the like terms to simplify the result: \n 21\(x^5\) - 28\(x^4\) -15\(x^4\) + 20\(x^3\) + 18\(x^3\) - 24\(x^2\) results in \n 21\(x^5\) -43\((x^4)\) + 38\((x^3)\) - 24\((x^2)\)

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