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What real number that should be added to the polynomial f(x) = 81x2 - 31, so that it is exactly divisible by 9x + 1?
  • a)
    40
  • b)
    10
  • c)
    30
  • d)
    20
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
What real number that should be added to the polynomial f(x) = 81x2- 3...
81x2 - 31 is exactly divisible by 9x + 1
Hence, on dividing 81x2 - 31 by 9x + 1
We get, 9x - 1 as quotient and remainder as -30.
So if we add 30 to 81x2 - 31, it will be exactly divisible by 9x + 1.
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Most Upvoted Answer
What real number that should be added to the polynomial f(x) = 81x2- 3...
81x2 - 31 is exactly divisible by 9x + 1
Hence, on dividing 81x2 - 31 by 9x + 1
We get, 9x - 1 as quotient and remainder as -30.
So if we add 30 to 81x2 - 31, it will be exactly divisible by 9x + 1.
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What real number that should be added to the polynomial f(x) = 81x2- 3...
To find the real number that should be added to the polynomial f(x) = 81x^2 - 31, so that it is exactly divisible by 9x - 1, we need to use the concept of polynomial division.

1. Polynomial Division:
Polynomial division is similar to long division, where we divide one polynomial by another polynomial. It helps us determine if one polynomial is a factor of another and find the quotient and remainder.

2. Divisibility Criterion:
For a polynomial to be exactly divisible by another polynomial, the remainder should be zero.

3. Formula:
To find the real number that should be added, we can set up the polynomial division equation as follows:
(81x^2 - 31) ÷ (9x - 1) = Q(x) + R(x)/(9x - 1)
where Q(x) represents the quotient and R(x) represents the remainder.

4. Setting up the Equation:
We want the remainder to be zero, so we set R(x) = 0:
(81x^2 - 31) ÷ (9x - 1) = Q(x) + 0/(9x - 1)
Simplifying further, we get:
(81x^2 - 31) ÷ (9x - 1) = Q(x)

5. Performing the Division:
Performing the polynomial division, we get:
Q(x) = 9x + 10

6. Interpretation:
The quotient Q(x) represents the polynomial that results from dividing f(x) by (9x - 1). The remainder is zero, indicating that (9x - 1) evenly divides f(x).

7. Finding the Real Number:
To find the real number that should be added, we set Q(x) equal to zero and solve for x:
9x + 10 = 0
9x = -10
x = -10/9

8. Conclusion:
The real number that should be added to the polynomial f(x) = 81x^2 - 31, so that it is exactly divisible by 9x - 1, is -10/9. Therefore, the correct answer is option C.
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Directions:Read the passage and choose the best answer to each question.PassagePeople use many different chemicals each day for common household tasks such as cleaning and food preparation.Since the inception of consumer protection laws, chemicals come with toxicity warning labels, directions about proper use, and cautions about the hazards of improper use. Some household chemicals can be quite dangerous, especially when mixed together. One such example is the reaction that occurs when mixing household bleach (NaOCl) with ammonia (NH3). The by-products of the reaction vary depending on the concentrations of the reactants. The following experiments were conducted to determine the levels at which certain by-products resulted from mixing bleach and ammonia.Experiment 1A known by-product of the reaction of bleach and ammonia is chlorine gas (Cl2). Chlorine gas has an intensely disagreeable suffocating odor, and is very poisonous. To determine the quantities of bleach and ammonia that, when mixed together, produce chlorine gas, a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each of the solutions; the amount added was gradually increased for each trial. The amount of chlorine gas produced in each trial was recorded and graphed in Figure 1.Experiment 2Another known by-product of the reaction of bleach and ammonia is nitrogen trichloride (NCl3). Nitrogen trichloride is a yellow, oily, pungent-smelling liquid, often found as a by-product of chemical reactions between nitrogen containing compounds and chlorine. It is highly explosive.To determine the quantities of bleach and ammonia that, when mixed together, produce NCl3, again a varying quantity of bleach was added to eight different ammonia–water solutions and the resulting NCl3 from each mixture was measured. A solution of 1.0 mole (mol) of NH3 in 1 kg of water was used in each trial. A certain quantity of NaOCl was added to each solution; the quantity addedwas gradually increased for each trial. The amount of nitrogen trichloride produced in each trial was recorded in see Table 1.Experiment 3In yet another reaction, bleach and ammonia combined under certain conditions produce a compound known as chloramine. Chloramine (NH2Cl) is a toxic substance commonly used in low concentrations as a disinfectant in municipal water systems as an alternative to chlorination.To determine the mixture of bleach and ammonia at which NH2Cl is produced, a varying amount of ammonia was added to eight different bleach–water solutions and the resulting chlorine gas from each mixture was collected and measured. A solution of 1.0 mole (mol) of NaOCl in 1 kg of water was used in each trial. A certain quantity of NH3 was added to each solution; the quantity of ammonia added was gradually increased for each trial. The amount of chloramine produced in each trial was recorded in Table 2.Q.The production of a certain plastic calls for a mixture of bleach and ammonia. However, the presence of chlorine gas is highly undesirable. Based on the results of Experiments 1, 2, and 3, which of the following specifications should be chosen?

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What real number that should be added to the polynomial f(x) = 81x2- 31, so that it is exactly divisible by 9x + 1?a)40b)10c)30d)20Correct answer is option 'C'. Can you explain this answer?
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