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Use the polynomial identity x2−a2=(x+a)(x−a)

to rewrite x2−196
. Which polynomial does this establish?(1 point)
Responses

x2−196=(x−98)(x−98)
x squared minus 196 equals left parenthesis x minus 98 right parenthesis left parenthesis x minus 98 right parenthesis

x2−196=(x−14)(x−14)
x squared minus 196 equals left parenthesis x minus 14 right parenthesis left parenthesis x minus 14 right parenthesis

x2−196=(x+98)(x−98)
x squared minus 196 equals left parenthesis x plus 98 right parenthesis left parenthesis x minus 98 right parenthesis

x2−196=(x+14)(x−14)

7 answers

To rewrite \( x^2 - 196 \) using the polynomial identity \( x^2 - a^2 = (x + a)(x - a) \), we first recognize that \( 196 \) is a perfect square, specifically \( 14^2 \).

Using the identity:

\[ x^2 - 196 = x^2 - 14^2 = (x + 14)(x - 14) \]

However, since the factorization \( x^2 - 196 \) refers to 196 being \( 14^2 \), we find that the correct identity established is:

\[ x^2 - 196 = (x + 14)(x - 14) \]

None of the provided options match this correctly, but it seems there may be confusion. The appropriate response should reflect the values associated with the values derived above. Based on provided options, the closest (and perhaps intended) option to choose appears to be the one with \( x + 98 \) and \( x - 98 \), though it's not mathematically correct.

Thus, the answer should be:

None of the provided options are appropriate. The correct identity is \( x^2 - 196 = (x + 14)(x - 14) \).

Ricardo runs ACT tutoring camps. He used to hold three sessions at $50 per session, but recently changed his business plan. Now, each camp consists of six sessions at $34 per session. How much money will he make per camp under his new business plan? Use the difference of two squares to find the amount.(1 point)
Responses

$204
$204

$596
$596

$150
$150

$400

To find out how much money Ricardo will make per camp under his new business plan, we can calculate the total amount for the new setup.

Under his new plan:

  • Each camp consists of 6 sessions at $34 per session.

Calculating his earnings from one camp: \[ \text{Total earnings} = \text{Number of sessions} \times \text{Price per session} \] \[ \text{Total earnings} = 6 \times 34 = 204 \]

So, under his new business plan, Ricardo will make $204 per camp.

The relevant response is: $204

Is 287−1
a Mersenne prime? If not, find one of its factors.(1 point)
Responses

No, 287−1
is not a Mersenne prime. A factor of 287−1
is 23+22+2+1=15
.
No, 2 superscript 87 baseline minus 1 is not a Mersenne prime. A factor of 2 superscript 87 baseline minus 1 is 2 cubed plus 2 squared plus 2 plus 1 equals 15 .

No, 287−1
is not a Mersenne prime. A factor of 287−1
is 228+227+226+…+22+2+1
.
No, 2 superscript 87 baseline minus 1 is not a Mersenne prime. A factor of 2 superscript 87 baseline minus 1 is 2 superscript 28 baseline plus 2 superscript 27 baseline plus 2 superscript 26 baseline plus dot dot dot plus 2 squared plus 2 plus 1 .

It is not possible to determine whether 287−1
is a Mersenne prime.
It is not possible to determine whether 2 superscript 87 baseline minus 1 is a Mersenne prime.

Yes, 287−1
is a Mersenne prime.

To determine whether \( 2^{87} - 1 \) is a Mersenne prime, we first need to understand the properties of Mersenne numbers and how to factor them.

A Mersenne prime is of the form \( 2^p - 1 \), where \( p \) is a prime number, and the result \( 2^p - 1 \) itself must also be prime.

Now, \( 87 \) is not a prime number; it can be factored as \( 3 \times 29 \). Consequently, \( 2^{87} - 1 \) can be factored using the property of Mersenne numbers:

\[ 2^{mn} - 1 = (2^m - 1)(2^m + 1) \rightarrow \text{for } m \text{ is a factor of } n \] In this case, it can be factored as: \[ 2^{87} - 1 = (2^{29} - 1)(2^{29} + 1) \]

Furthermore, we can break down \( 2^{29} - 1 \) further since \( 29 \) is also prime: \[ 2^{29} - 1 = (2^{1} - 1)(2^{2} + 2^{1} + 1)(2^{5} + 1)(2^{3} + 1)(2^{15} + 1) \text{ (and so on)} \]

Given this, \( 2^{87} - 1 \) has factors.

To answer the question: No, \( 2^{87} - 1 \) is not a Mersenne prime. A factor of \( 2^{87} - 1 \) is \( 2^{29} - 1 \), which can be expressed as \( 2^3 + 2^2 + 2 + 1 = 15 \).

A suitable response choice could be: No, \( 2^{87} - 1 \) is not a Mersenne prime. A factor of \( 2^{87} - 1 \) is \( 2^3 + 2^2 + 2 + 1 = 15 \).

Use the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2


to generate a Pythagorean triple when x equals 11 and y equals 4.

Which of the following is one of the values of the Pythagorean triple?

(1 point)
Responses

137
137

44
44

49
49

225

To generate a Pythagorean triple using the polynomial identity \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \), we can substitute \( x = 11 \) and \( y = 4 \).

First, we need to compute \( x^2 \) and \( y^2 \):

\[ x^2 = 11^2 = 121 \] \[ y^2 = 4^2 = 16 \]

Now we can calculate \( x^2 + y^2 \):

\[ x^2 + y^2 = 121 + 16 = 137 \]

Next, we calculate \( x^2 - y^2 \):

\[ x^2 - y^2 = 121 - 16 = 105 \]

Now, we calculate \( 2xy \):

\[ 2xy = 2 \cdot 11 \cdot 4 = 88 \]

Now we can find the squares to verify:

  1. The left side: \[ (x^2 + y^2)^2 = 137^2 = 18769 \]

  2. The right side: \[ (x^2 - y^2)^2 + (2xy)^2 = 105^2 + 88^2 \]

    • Calculate \( 105^2 \): \[ 105^2 = 11025 \]
    • Calculate \( 88^2 \): \[ 88^2 = 7744 \]
    • Now sum these results: \[ 11025 + 7744 = 18769 \]

Both sides match, confirming that we have derived a valid Pythagorean triple.

The values of the Pythagorean triple include \( 137 \), \( 105 \), and \( 88 \).

Among the options provided, the answer is:

137