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3 years ago

How do I factor the following expression?

Mathematics

2 answers:

Crazy boy [7]3 years ago

6 0

$\bf x^6-9x^4-81x^2+729\implies \stackrel{\textit{let's do some grouping}}{(x^6-9x^4)-(81x^2-729)} \\\\\\ \stackrel{\textit{some common factoring}}{x^4\boxed{(x^2-9)}-81\boxed{(x^2-9)}}\implies \stackrel{\textit{common factoring}}{\boxed{(x^2-9)}(x^4-81)} \\\\\\ \stackrel{\textit{difference of squares}}{[x^2-3^2][(x^2)^2-(3^2)^2]}\implies (x-3)(x+3)\stackrel{\textit{difference of squares}}{(x^2-3^2)}(x^2+3^2) \\\\\\ (x-3)(x+3)~(x-3)(x+3)~(x^2+9)$

amid [387]3 years ago

6 0

On a hunch, I decided to check whether 729 is a 6th power, and found that it is:

729^(1/6) = 3.

Next, I decided to divide x-3 into x^6 + 0x^5 - 9x^4 + 0x^3 - 81x^2 + 0x + 729, through synthetic division and using 3 as my divisor. This left no remainder. The coefficients of the quotient were as follows: 3 3 0 0 -81 -243, which represents:

3x^5 + 3x^4 + 0x^3 + 0x^2 -81x -243

Next, I applied "factoring by grouping:"

3x^4(x+1) -81(x+3) = (x+3)(3x^4 - 81).

Note that 3x^4 - 81 is the same as 3(x^4-27).

Thus, the original polynomial factors into (x-3)(3)(x^4 - 27).

You might be interested in

The vertex of this parabola is at (2, -4). When the y-value is -3, the x-value is -3. What is the coefficient of the squared ter

lyudmila [28]

Answer:

The squared term is 1/25

Step-by-step explanation:

The parabola equation in the vertex form is

$y = a \times (x - h)^2 + k$

where point (h,k) is the vertex and a is the squared term. In this case h = 2 and k = -4. On the other hand, we know that y = -3 and x = -3 are in the parabola. Replacing these values in the formula gives

$-3 = a \times (-3 - 2)^2 - 4$

Solving for a

$-3 + 4 = a \times (-5)^2$

$1 = a \times 25$

$a = \frac{1}{25}$

6 0

3 years ago

Find the set of outcomes that occur when we flip 3 coins and obtain at least two tails.

Vikentia [17]

First, I would show all the outcomes possible. To determine the number of the total outcomes, you use the formula: rⁿ, where r is the number of outcomes in 1 flip, and n is the number of lips. Thus, 2³ = 8 outcomes. These outcomes are:

1. HHH
2. TTT
3. HHT
<em>4. HTT</em>
5. HTH
6. THH
<em>7. TTH</em>
<em>8. THT</em>

From the given outcomes, there are <em>3 outcomes</em> that has at least two tails. These are the ones in bold characters.

5 0

3 years ago

Find four distinct complex numbers (which are neither purely imaginary nor purely real) such that each has an absolute value of

Luda [366]

Answer:

0.5 + 2.985i
1 + 2.828i
1.5 + 2.598i
2 + 2.236i

Explanation:

Complex numbers have the general form a + bi, where a is the real part and b is the imaginary part.

Since, the numbers are neither purely imaginary nor purely real a ≠ 0 and b ≠ 0.

The absolute value of a complex number is its distance to the origin (0,0), so you use Pythagorean theorem to calculate the absolute value. Calling it |C|, that is:

$|C| = \sqrt{a^2+b^2}$

Then, the work consists in finding pairs (a,b) for which:

$\sqrt{a^2+b^2}=3$

You can do it by setting any arbitrary value less than 3 to a or b and solving for the other:

$\sqrt{a^2+b^2}=3\\ \\ a^2+b^2=3^2\\ \\ a^2=9-b^2\\ \\ a=\sqrt{9-b^2}$

I will use b =0.5, b = 1, b = 1.5, b = 2

$b=0.5;a=\sqrt{9-0.5^2}=2.958\\ \\b=1;a=\sqrt{9-1^2}=2.828\\ \\b=1.5;a=\sqrt{9-1.5^2}=2.598\\ \\b=2;a=\sqrt{9-2^2}=2.236$

Then, four distinct complex numbers that have an absolute value of 3 are:

0.5 + 2.985i
1 + 2.828i
1.5 + 2.598i
2 + 2.236i

4 0

3 years ago

Which inequality is true? 1.5 greater-than 2 and one-half

Travka [436]

The inequality negative 3 and one-half greater-than negative 4.5 or -3.5 > -4.5 is true.

<h3>What is inequality?</h3>

It is defined as the expression in mathematics in which both sides are not equal they have mathematical signs either less than or greater than known as inequality.

We have inequalities given:

$1.5 > 2\dfrac{1}{2}$