Algebra

The second-degree polynomial function ( f(x) ) with a lead coefficient of 3 and roots 4 and 1 can be expressed using the factored form: ( f(x) = a(x - r_1)(x - r_2) ), where ( a ) is the lead coefficient, and ( r_1 ) and ( r_2 ) are the roots. Substituting the values, we have ( f(x) = 3(x - 4)(x - 1) ). Expanding this gives ( f(x) = 3(x^2 - 5x + 4) = 3x^2 - 15x + 12 ). Thus, the polynomial is ( f(x) = 3x^2 - 15x + 12 ).

The numerical value of the slope indicates how steep a line is and the direction it slants. A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. The greater the absolute value of the slope, the steeper the line; for example, a slope of 3 is steeper than a slope of 1. A slope of zero represents a horizontal line, while an undefined slope corresponds to a vertical line.

In a linear optimization graph, two optimal solutions can occur when the objective function line is parallel to a constraint line at the vertex of the feasible region. For example, consider a feasible region defined by constraints forming a polygon with one edge being a constraint line. If the objective function is maximized along that edge, any point along that edge within the feasible region will yield the same optimal value, resulting in multiple optimal solutions.

The result of solving an equation to find values for the variables is known as the solution set. This set includes all possible values that satisfy the equation, making it true when substituted back into the original equation. If there is a unique solution, it is a single value; if there are multiple solutions, they are typically expressed in a set or as a range. In some cases, there may be no solution at all.

5x^(2) + 2x - 4 = 2x^(2)

Hence

3x^(2) + 2x - 4 = 0

Now apply the Quadratic Equation.

x = { - 2 +/- sqrt[2^(2) - 4(3)(-4)]} / 2(3)

x = {-2 +/- sqrt[4 + 48]} / 6

x = { -2 +/- sqrt(50_] / 6

x = { -2 +/-5sqrt(2)} / 6

x = (-2 - 5sqrt(2))/ 6

&

x = (-2 + 5sqrt(2))/ 6

Since the square root of '2' is an Irrational Number. (decimals go to infinity, the answer is left in 'surd' form .

In one word 'NO'

Since it is and 'x^(2)' term, this will make a parabolic curve, not linear (straight line).

5x^(2) = 11

Divide both saudes by 5'

Hence

x^(2) = 11/5

Square root both sides

x = +/-sqrt (11/5)

Or x = +/-sqrt(2.2)

This would normally by left in 'surd' form, because the answer is an IRRATIONAL number; the decimals go to inifinity!!!!!

However, per calculator

x = +/- 1.483239697.....

5x^(2) + 15x = 0

Factor

Factor our '5x'

Hence

5x(x - 3) = 0

Hence it follows that

5x = 0

Therefore x = 0

or the other multiplicand

x - 3 = 0

x = 3

So the answer is x = 0 or x = 3 (two answers).

15x = 360

Algebrauically rearrange

x = 360/15

Cancel down by '5'

x = 72/3

Cancel down by '3'

x = 24/1

x = 24 The answer!!!!!

0.25 X 360 = 90

To mulyply decimals by long multiplication.

360

x 25 ( NB we have temporarily dropped the decimal point).

7200 (360 x 20)

1800 (360 x 5)

9000

=====

We note that there were only 2 decimals places in the multiplicands. So the answer has 2 decimal places.

Hence 9000 becomes , 90,00 or just plain '90'.

Another way is to note that ' 0.25 = 1/4'

So again multiply 360 x 1/4

Multiplication of fractions.

360/1 x 1/4 =

Cancel down by '4'

90/1 X 1/1 = 90/1 = 90

Another 'Short Circuit' method. is to note that 9 x 4 = 36

Hence 90 x 4 = 360

So 360/ 4 - 360 x 1/4 = 90 .

Careful with this last method, 'Short Circuits; can be dangerous, mabd so you my end up with the wrong answer.

By long multiplication

90

x 12

900

180

1080

===== Done!!!!!

Appears we are adding 6 each time. So the nth term is a_(n-1)+6 where n-1 is a subscript of a.

rDNA, or ribosomal DNA, encodes the RNA components of ribosomes, which are essential for protein synthesis in all living cells. It is involved in the production of ribosomal RNA (rRNA), a critical component of ribosomes, and plays a key role in the assembly of ribosomal subunits. Additionally, rDNA is often used as a molecular marker in genetic studies due to its repetitive nature and evolutionary significance.

To reflect the graph of ( f(x) = x - 1 ) across the y-axis, you replace ( x ) with ( -x ), resulting in the equation ( f(-x) = -x - 1 ). To translate this graph 4 units down, you subtract 4 from the entire function, giving you ( f(-x) - 4 = -x - 1 - 4 = -x - 5 ). Thus, the final transformed function is ( f(-x) - 4 = -x - 5 ).

The graph of the inequality (4x + 3y \leq 24) represents a region in the coordinate plane. First, you would graph the line (4x + 3y = 24) by finding the x-intercept (6,0) and y-intercept (0,8). The area below this line, including the line itself, is the solution to the inequality, indicating all the points ((x, y)) that satisfy (4x + 3y) being less than or equal to 24.

To find the domain of the function ( f(x) = x^2 + 1 ), we identify the set of all possible input values for ( x ). Since this is a polynomial function, the domain is all real numbers, expressed as ( (-\infty, \infty) ). The range is determined by analyzing the output values; the minimum value of ( f(x) ) occurs at ( x = 0 ), giving ( f(0) = 1 ). Therefore, the range is ( [1, \infty) ).

It seems there might be a typo or confusion in your question. If you meant to say "Evaluate 67s for s = 8," then substituting s with 8 gives you 67 * 8 = 536. If "67s 3" refers to something else, please clarify for more accurate assistance.

8.99 multiplied by 4 equals 35.96. You can calculate this by multiplying the two numbers directly.

The expected value of a random variable ( x ) is a measure of the central tendency and is calculated as the weighted average of all possible values, where each value is weighted by its probability of occurrence. Mathematically, it is expressed as ( E(x) = \sum (x_i \cdot P(x_i)) ) for discrete variables, or as ( E(x) = \int x \cdot f(x) , dx ) for continuous variables, where ( f(x) ) is the probability density function. The expected value provides insight into the long-term average outcome of a random variable in a probability distribution.

Nematocysts are specialized stinging cells found in cnidarians, such as jellyfish, sea anemones, and corals. Their primary function is to capture prey and provide defense against predators. When triggered, nematocysts release a coiled thread that can inject toxins into the target, immobilizing or harming it. This mechanism is essential for the survival and feeding of these marine organisms.

The identity (\cos(x) = \sin(90^\circ - x)) arises from the co-function relationship in trigonometry. For angles in the range (0^\circ \leq x \leq 90^\circ), the sine of an angle is equal to the cosine of its complementary angle. Thus, when you replace (90^\circ - x) with (x) in the sine function, you obtain (\cos(x)). This relationship highlights the symmetry of the sine and cosine functions in a right triangle.

To determine if the line of best fit is appropriate for the data, examine the residual plot for randomness. If the residuals are randomly scattered around the horizontal axis without any discernible pattern, it suggests that the linear model is suitable. Conversely, if the residuals display a pattern (such as a curve), it indicates that a linear model may not be the best fit for the data.

16 x 2 equals 32. This is a simple multiplication problem where you double the number 16. Therefore, the final answer is 32.

To find the quotient of a binomial or polynomial when there is a remainder, perform polynomial long division or synthetic division. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient. Multiply the entire divisor by this term and subtract the result from the dividend, bringing down the next term as needed. Continue this process until you reach a remainder that is of lower degree than the divisor, which can be expressed as ( \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}} ).

When a point with coordinates ((x, y)) is reflected over the x-axis, its x-coordinate remains the same while the y-coordinate changes sign. Thus, the new coordinates of the reflected point become ((x, -y)). This transformation effectively flips the point vertically, moving it to the opposite side of the x-axis.