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Chapter 0: Problem 85

Perform the indicated operation or operations. $$(5 x-7)(3 x-2)-(4 x-5)(6 x-1)$$

Short Answer

Expert verified
The simplified expression is \( -9 x^2 + 3 x + 9 \).

Step by step solution

01

Apply the Distributive Property

First, it's necessary to multiply the terms in the binomial expressions. Using the distributive property, it is \( (5 x * 3 x) - (5 x * 2) - (7 * 3 x) + (7 * 2) \) for the first binomial and \( (4 x * 6 x) - (4 x * 1) - (5 * 6 x) + (5 * 1) \) for the second binomial.
02

Simplify each binomial product

After performing the multiplication operations, you'll get two polynomial expressions. For the first binomial, it is \( 15 x^2 - 10 x - 21 x + 14 \) and for the second binomial, it is \( 24 x^2 - 4 x - 30 x + 5 \),
03

Combine Like Terms

Next, combine like terms within each polynomial. This will give \( 15 x^2 -31 x +14 \) for the first polynomial and \( 24 x^2 - 34 x + 5 \) for the second polynomial.
04

Subtract the second polynomial from the first

The last operation is subtracting the second polynomial from the first, which gives \( (15 x^2 -31 x + 14) - (24 x^2 - 34 x + 5) \). Remember that subtracting a polynomial is the same as adding its opposite. Therefore, all the signs of the terms in the second polynomial are changed. The expression becomes: \( 15 x^2 -31 x +14 - 24 x^2 + 34 x - 5 \).
05

Combine Like Terms

Finally, combine like terms in the resulting polynomial. This yields the simplified expression \( (-9 x^2 + 3 x + 9) \).

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Most popular questions from this chapter

Perform the indicated operations. $$\frac{1}{x^{n}-1}-\frac{1}{x^{n}+1}-\frac{1}{x^{2 n}-1}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I prefer interval notation over set-builder notation because it takes less space to write solution sets.

The rational expression $$\frac{130 x}{100-x}$$ describes the cost, in millions of dollars, to inoculate \(x\) percent of the population against a particular strain of flu. a. Evaluate the expression for \(x=40, x=80,\) and \(x=90\) Describe the meaning of each evaluation in terms of percentage inoculated and cost. b. For what value of \(x\) is the expression undefined? c. What happens to the cost as \(x\) approaches \(100 \% ?\) How can you interpret this observation?

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

This will help you prepare for the material covered in the next section. Evaluate $$\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$$ for \(a=2, b=9,\) and \(c=-5\)
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