A polynomial expression can be factored by grouping when it has at least four terms. To factor by grouping, group the terms into pairs, factor out the greatest common factor from each pair, apply the distributive property to factor out the common binomial factor, and simplify.
Explanation:A polynomial expression can be factored by grouping when it has at least four terms. To factor by grouping, follow these steps:
Group the terms of the polynomial into pairs.Factor out the greatest common factor from each pair of terms.Apply the distributive property to factor out the common binomial factor.Simplify the expression by combining like terms.For example, let's consider the polynomial expression 3x^3 - 3x^2 + 2x - 2. We can group the terms as (3x^3 - 3x^2) + (2x - 2). Factor out the greatest common factor from each pair of terms, which gives us 3x^2(x - 1) + 2(x - 1). Applying the distributive property, we can factor out the common binomial factor (x - 1), resulting in (x - 1)(3x^2 + 2). This is the factored form of the original polynomial.
A polynomial expression that can be factored by grouping typically has four terms that are grouped into pairs, with each pair having a common factor. After factoring out the common factors from each pair, if the resulting binomials are identical, the expression can then be factored into the product of two binomials.
Explanation:The structure of a polynomial expression that can be factored by grouping typically involves four terms with the possibility of factoring pairs of terms separately. To use this method, you would look for common factors in the first two terms and in the last two terms. If a common factor is found in both pairs, you can then factor out these common factors and check if the resulting binomials are identical. If so, you can factor out the binomial, leaving you with a product of two binomials as the factored form of the polynomial.
Here is a step-by-step example:
Consider the polynomial ax + ay + bx + by.Group the first two terms and the last two terms: (ax + ay) + (bx + by).Factor out the common factors in each group: a(x + y) + b(x + y).Notice that the binomial (x + y) is common to both groups, so you can factor it out: (x + y)(a + b).Thus, you've factored the original polynomial by grouping.
Which of the following is unnecessary when multiplying these polynomials?
(2x^3+2x^2-3x)(9x^2-4x+5)
A)use of the distributive property
B)use of the multiplication property of exponents
C)combine like terms
D)find the GCF
Subtract 11a2+42a-6 from 14a2-24a+7
The values of subtraction of 11a²+42a-6 from 14a²-24a+7 is
3a² - 66a + 1.
What is Expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)
We have to Subtract 11a²+42a-6 from 14a²-24a+7.
So, the subtraction is as follows
= 14a²-24a+7 - ( 11a²+42a-6)
= 14a²-24a+7 - 11a² - 42a + 6
= 14a²- 11a² -24a - 42a + 7- 6
= 3a² - 66a + 1
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Andrea and raleigh are each rolling a fair, six-sided die. they roll their dice simultaneously, individually keeping a sum until someone reaches 100; whoever reaches 100 first wins. (if they reach 100 on the same roll, it's a tie.) andrea's die has sides 1, 2, 3, 4, 5, and 6. raleigh's has sides 1, 1, 1, 6, 6, and 6. who is more likely to win?
Let An be the likely number of rolls for Andrea to arrive at 100 or more, given that she is n away from the target of 100 (which means her current total is 100 - n ). Then we get the recurrence relation
An = 1 + (1/6) (An-1 + An-2 + An-3 + An-4 + An-5 + An – 6)
and the initial conditions A0 = A-1 = A-2 = A-3 = A-4 = A-5 = 0. This results in A100 = 29.0476 as the amount of expected rolls that Andrea has to make.
Now let Rn to be expected amount of rolls for Raleigh to arrive at 100 or more, given than he is also n away from the target of 100. Now the recurrence relation is:
Rn = 1 + (1/2) (Rn-1 + Rn-6)
with the initial conditions R0 = R-1 = R-2 = R-3 = R-4 = R-5 = 0. This gives us R100 = 29.1837 as the expected amount of rolls for Raleigh.
Therefore Andrea has to make less number of rolls thus she is expected to win.
Winner: Andrea
Match the following items by evaluating the expression for x=-3.
x^-3 1/9,-3,-1/3,9,1
x^-1 -1/3,1,9,1/9,-3
x^0 -3,9,-1/3,1/9,1
x^1 1/9,-3,-1/3,1,9
x^2 -3,1,9,1/9,-1/3
Answer:
The values of given expression at x=-3 are [tex]x^{-3}= -\frac{1}{27}[/tex], [tex]x^{-1}=-\frac{1}{3}[/tex], [tex]x^{0}=1[/tex], [tex]x^{1}=-3[/tex] and [tex]x^{2}=9[/tex].
Step-by-step explanation:
We need to find the value of each expression at x=-3.
The given expression is
[tex]x^{-3}[/tex]
It can be written as
[tex]x^{-3}=\frac{1}{x^3}[/tex] [tex][\because x^{-n}=\frac{1}{x^n}][/tex]
Substitute x=-3 in the above expression.
[tex]x^{-3}=\frac{1}{(-3)^3}\Rightarrow -\frac{1}{27}[/tex]
Similarly,
[tex]x^{-1}=\frac{1}{x}\Rightarrow -\frac{1}{3}[/tex]
[tex]x^{0}=1[/tex] [tex][\because x^{0}=1][/tex]
[tex]x^{1}=(-3)^1=-3[/tex]
[tex]x^{2}=(-3)^2=9[/tex]
Therefore the values of given expression at x=-3 are [tex]x^{-3}= -\frac{1}{27}[/tex], [tex]x^{-1}=-\frac{1}{3}[/tex], [tex]x^{0}=1[/tex], [tex]x^{1}=-3[/tex] and [tex]x^{2}=9[/tex].
What is the solution to the system of equations graphed below?
a. (2,-3)
b. (-3,2)
c. (-2,3)
d. (3,-2)
Two integers that vary in sign
Simplify (−34.67)0. I'm not really understanding how to do this so please someone help me.
Bill is riding his bicycle. He takes 5 hours to ride 62.5 kilometers. What is his speed?
Solve the quadratic equation -x^2=4
Find the value of y. Look at image attached
The profit in manufacturing x refrigerators per day, is given by the profit relation p = ¡3x2 + 240x ¡ 800 dollars. a how many refrigerators should be made each day to maximise the total profit? b what is the maximum profit?
Final answer:
The firm should produce 40 refrigerators per day to maximize profit, resulting in a maximum profit of $4,000. A flat tax would reduce profit without affecting price directly, while a per unit tax could increase prices and alter the profit-maximizing output and profit levels.
Explanation:
Maximizing Profit and Effects of Taxes on a Firm's Profit
The profit that a firm makes from manufacturing x refrigerators per day is given by the profit function p = -3x^2 + 240x - 800, where p stands for profit. To maximize the total profit, we need to find the vertex of this quadratic function, which represents the highest point of the parabola on a graph. The x-coordinate of the vertex can be found using the formula -b/(2a), where a is the coefficient of x^2 and b is the coefficient of x. Substituting the values from the profit function, we get x = -240/(2*(-3)), which simplifies to x = 40. Thus, the firm should produce 40 refrigerators per day to maximize profit.
To find the maximum profit, we substitute x = 40 back into the profit function: p = -3(40)^2 + 240(40) - 800. After calculating, the maximum profit comes out to be $4,000 per day.
If a tax of $1,000 per day is imposed on the firm, this would lower the profit by the amount of the tax, without directly affecting the price. It's uncertain how the firm might react—whether they decide to increase the price to compensate or not is a strategic decision.
A tax of $100 per unit would likely cause the firm to increase the price by at least $100 to maintain their profit margins. The new profit function would be p = -3x^2 + (240-100)x - 800, which would also affect the profit-maximizing output. After recalculating with the new function, the firm would adjust its output and profit accordingly.
How many terms are in the polynomial abcd + e - h 2? two three five six
Answer:
The number of terms in the polynomial are:
Three.
Step-by-step explanation:
We are given a polynomial expression in terms of variable a,b,c,d , e and h as:
[tex]abcd+e-h^2[/tex]
We know that in any algebraic expression the number of terms are the number of unlike expressions that are separated by a minus or plus i.e. addition sign.
Hence here by looking at our polynomial expression we see that there are three terms in the expression :
1) [tex]abcd[/tex]
2) [tex]e[/tex]
3) [tex]h^2[/tex]
When constructing inscribed polygons how can you be sure the figure inscribed is a regular polygon
The cost of producing x soccer balls in thousands of dollars is represented by h(x) = 5x + 6. The revenue is represented by k(x) = 9x – 2. Which expression represents the profit, (k – h)(x), of producing soccer balls?
Which transformations can be used to map a triangle with vertices A(2, 2), B(4, 1), C(4, 5) to A’(–2, –2), B’(–1, –4), C’(–5, –4)? a 180 rotation about the origin a 90 counterclockwise rotation about the origin and a translation down 4 units a 90 clockwise rotation about the origin and a reflection over the y-axis a reflection over the y-axis and then a 90 clockwise rotation about the origin
Answer:
C. a 90 clockwise rotation about the origin and a reflection over the y-axis
Step-by-step explanation:
Noa drove from the Dead Sea up to Jerusalem, and her altitude increased at a constant rate of 740 meters per hour. When she arrived in Jerusalem after 1.5 hours of driving, her altitude was 710 meters above sea level.
Let A(t) denote Noa's altitude relative to sea level A (measured in meters) as a function of time t (measured in hours).
She stared at -400 meters below sea level and the equation is A(t)=740t-400
Functions and valuesFunctions are written in terms of variables
From the given question;
Let a₀ is the initial height or the height she started at
Required
Initial height
After 1.5 hours, she was 710 meters above sea level
Substitute the given time into the function
A(1.5)=740(1.5)+a₀=710
740(1.5)+a₀=710
1110+a₀=710
a₀=-400
This shows that she stared at -400 meters below sea level and the equation is A(t)=740t-400
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Three times the greater of two consecutive odd integers is five less than four times the smaller. find the two numbers.
Point B is between A and C. If AB = 6 cm and BC = 11 cm, what is AC?
If the measure of angle is 38 what is the measure of its complement
On an up hill hike Ted climbs at 3 mph. Going back down, he runs at 5 mph. If it takes him forty minutes longer to climb up than run down, then what is the length of the hike?
Answer: He hiked 10 miles.
Step-by-step explanation:
3mi/hr x 1 2/3 hr = 5 miles. 5 mi/hr x 1 hr = 5 miles.
Which statement best describes the graph of x3 − 3x2 − x + 3?
It starts down on the left and goes up on the right and intersects the x-axis at x = −1, 2, and 3.
It starts down on the left and goes up on the right and intersects the x-axis at x = −1, 1, and 3.
It starts up on the left and goes down on the right and intersects the x-axis at x = −1, 2, and 3.
It starts up on the left and goes down on the right and intersects the x-axis at x = −1, 1, and 3.
It starts down on the left and goes up on the right and intersects the x-axis at x = −1, 1, and 3.
Which statement best describes the graph?
We want to describe the graph of:
[tex]y = x^3 - 3x^2 - x + 3[/tex]
We can see that the y-intercept is y = 3, and the x-intercepts can be seen in the graph below, these are at: x = -1, x = 1 and x = 3.
We also can see that it is cubic, so it starts down.
Then the correct option is the second one:
"It starts down on the left and goes up on the right and intersects the x-axis at x = −1, 1, and 3."
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which figure has both line of symmetry and rotational symmetry
Answer:
A line of symmetry is when you can put a line through a shape and you can fold it and it is identical on both sides, and rotational is when you can rotate it and at some point it is in the same spot as it was before, so yes, you are right. the octogon.
Step-by-step explanation:
(02.06 LC)
The figure below shows parallel lines cut by a transversal:
Which statement is true about ∠1 and ∠2?
A. ∠1 and ∠2 are complementary because they are a pair of corresponding angles.
B. ∠1 and ∠2 are congruent because they are pair of corresponding angles
C.∠1 and ∠2 are complementary because they are a pair of alternate interior angles.
D.∠1 and ∠2 are congruent because they are pair of alternate interior angles.
For the following geometric sequence, find the recursive formula.
{-80, 20, -5, ...}
Answer:
[tex]x_{n+1}=-\frac{x_{n}}{4}[/tex]
Step-by-step explanation:
We can get geometric sequence dividing each term by -4
So we will have:
[tex]x_{n+1}=\frac{x_{n}}{-4}=-\frac{x_{n}}{4}[/tex]
We can prove it putting the first value in this recursive equation:
x₁=-80
n = 1
[tex]x_{2}=-\frac{x_{1}}{4}=-\frac{-80}{4}=20[/tex]
If x₂ = 20, the next value will be:
[tex]x_{3}=-\frac{x_{2}}{4}=-\frac{20}{4}=-5[/tex]
I hope it helps you!
If only 4.5% of the non- historically protected have a field stone and mortar foundation, how many homes is this?
Shape 1 and shape 2 are plotted on a coordinate plane. Which statement about the shapes is true?
in a batch of 280 water purifiers 12 were found to be defective. What is the probability that a water purifier chosen at random will be defective? Write the probability as a percent. Round to the nearest tenth of a percent if necessary. A. 4.3% b. 5.6% c. 95.7% d. 71.8%
the answer would be 4.3 percent
Find the zeros of the function. F(x)= cube root of x
Carly and janiya put some money into their money boxes every week. the amounts of money (y), in dollars, in their money boxes after certain amounts of time (x), in weeks, are shown by the equations below:carlyy = 60x + 40janiyay = 50x + 80after how many weeks will carly and janiya have the same amount of money in their money boxes and what will that amount be? 5 weeks, $280 5 weeks, $340 4 weeks, $280 4 weeks, $340
Which of the following questions describes the equation w + 9 = -17? What number, A. when decreased by nine, is equal to negative seventeen?
B. What number, when added to negative seventeen, equals nine?
C. What number, when increased by nine, results in negative seventeen?
D. What number, when subtracted from nine, equals negative seventeen?