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Chapter 0: Problem 14

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(8 x^{2}+7 x-5\right)-\left(3 x^{2}-4 x\right)-\left(-6 x^{3}-5 x^{2}+3\right)$$

Short Answer

Expert verified
The resulting polynomial in standard form is \(-6x^{3}+10x^{2}+11x-2\) and its degree is 3.

Step by step solution

01

Distributing Negatives

First, distribute the negative sign (where necessary) to each term inside the parentheses. This gives: \(8x^{2}+7x-5 - 3x^{2}+4x - (-6x^{3}+5x^{2}-3)\).
02

Simplify the Expression

Next, simplify the expression by combining like terms, it becomes: \(-6x^{3}+(8-3+5)x^{2}+(7+4)x-5+3\). This simplifies to \(-6x^{3}+10x^{2}+11x-2\).
03

Write in Standard Form and Indicate Degree

Finally, rearrange the resulting polynomial in standard form and indicate its degree, that is: \(-6x^{3}+10x^{2}+11x-2\). The degree of the polynomial is 3, since the maximum exponent of x is 3.

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Most popular questions from this chapter

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

Evaluate each expression without using a calculator. $$125^{\frac{2}{3}}$$

The early Greeks believed that the most pleasing of all rectangles were golden rectangles, whose ratio of width to height is $$\frac{w}{h}=\frac{2}{\sqrt{5}-1}$$ The Parthenon at Athens fits into a golden rectangle once the triangular pediment is reconstructed. (IMAGE CANT COPY) Rationalize the denominator of the golden ratio. Then use a calculator and find the ratio of width to height, correct to the nearest hundredth, in golden rectangles.

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Evaluate each expression without using a calculator. $$8^{\frac{2}{3}}$$
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