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Chapter 5: Problem 95

Perform the indicated operations. $$4 x^{2}\left(5 x^{3}+3 x-2\right)-5 x^{3}\left(x^{2}-6\right)$$

Short Answer

Expert verified
The simplified expression is \(15x^5 + 42x^3 - 8x^2\)

Step by step solution

01

Distribute \(4x^2\) and \(-5x^3\)

Distribute \(4x^2\) to every term in the parentheses \((5x^3+3x-2)\) and \(-5x^3\) to every term in \((x^2-6)\). We have: \[4x^2 * 5x^3 = 20x^{2+3} = 20x^5\]\[4x^2 * 3x = 12x^{2+1} = 12x^3\]\[4x^2 * -2 = -8x^2\]\[-5x^3 * x^2 = -5x^{3+2} = -5x^5\]\[-5x^3 * -6 = 30x^3\]
02

Write down the simplified expression

The simplified expression after distributing the terms is:\[20x^5 + 12x^3 - 8x^2 - 5x^5 + 30x^3\]
03

Combine like terms

Next, the like terms in the simplified expression are combined. This involves adding or subtracting the coefficients (numbers in front) of like terms. Like terms are terms that have the same variable and exponent.Adding the \(x^5\)-terms together: \(20x^5 - 5x^5 = 15x^5\),Adding the \(x^3\)-terms together: \(12x^3 + 30x^3 = 42x^3\),The \(x^2\)-term remains the same as it doesn't have any like term: \(-8x^2\).
04

Write the final expression

By rearranging and writing the final expression in descending order of the exponents, the final simplified expression becomes:\[15x^5 + 42x^3 - 8x^2\]

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Most popular questions from this chapter

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

Perform the indicated computations. Express answers in scientific notation. $$\frac{\left(1.2 \times 10^{6}\right)\left(8.7 \times 10^{-2}\right)}{\left(2.9 \times 10^{6}\right)\left(3 \times 10^{-3}\right)}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. If \(5^{-2}\) is raised to the third power, the result is a number between 0 and 1

Use a vertical format to find each product. $$\begin{array}{r}9 y^{3}-7 y^{2}+5 y \\\3 y^{2}+5 y \\\\\hline\end{array}$$

Simplify: \(24+8 \cdot 3+28 \div(-7)\).
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