Ask a New Question
Menu

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

to generate a Pythagorean triple when x equals 10 and y equals 6. Which of the following is one of the values of the Pythagorean triple?(1 point)
Responses

256
256

16
16

60
60

120

7 answers

To generate a Pythagorean triple using the given polynomial identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\) with \(x = 10\) and \(y = 6\), we first calculate \(x^2\) and \(y^2\):

\[ x^2 = 10^2 = 100 \] \[ y^2 = 6^2 = 36 \]

Now we can compute each component of the identity.

  1. Calculate \(x^2 + y^2\):

\[ x^2 + y^2 = 100 + 36 = 136 \]

  1. Calculate \(x^2 - y^2\):

\[ x^2 - y^2 = 100 - 36 = 64 \]

  1. Calculate \(2xy\):

\[ 2xy = 2 \cdot 10 \cdot 6 = 120 \]

Now, we can substitute these values into the identity:

\[ (x^2 + y^2)^2 = 136^2 \] \[ (x^2 - y^2)^2 + (2xy)^2 = 64^2 + 120^2 \]

Now we can calculate \(136^2\), \(64^2\), and \(120^2\):

\[ 136^2 = 18496 \] \[ 64^2 = 4096 \] \[ 120^2 = 14400 \]

Now we can verify the identity:

\[ 64^2 + 120^2 = 4096 + 14400 = 18496 \]

The identity holds true.

The Pythagorean triple generated from the values we computed is:

  • One of the values is \(120\) (from \(2xy\)).

Thus, the correct answer is \(120\).

Samanta’s parents took her to a theme park and told her that she could pick out three toys from the gift shop. If there are nine different toys in total, how many different selections of three toys could she make? Write out Pascal’s Triangle from row 0 to row 9 and use it to solve the problem.

To determine how many different selections of three toys Samantha could make from a total of nine different toys, we need to compute the combination \( \binom{9}{3} \), which represents the number of ways to choose 3 items from 9 without regard to the order of selection.

Using the formula for combinations, we have:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In this case, \( n = 9 \) and \( r = 3 \):

\[ \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3! \cdot 6!} \]

Now, we can simplify:

\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84 \]

Thus, there are 84 different selections of three toys.

Pascal's Triangle

To verify our result using Pascal's Triangle, we can construct it up to row 9:

Row 0:                  1
Row 1:                1   1
Row 2:              1   2   1
Row 3:            1   3   3   1
Row 4:          1   4   6   4   1
Row 5:        1   5  10  10   5   1
Row 6:      1   6  15  20  15   6   1
Row 7:    1   7  21  35  21   7   1
Row 8:  1   8  28  56  70  28   8   1
Row 9: 1   9  36  84  126  36   9   1

We see that the value \( \binom{9}{3} \) appears in Row 9, Column 3, which is \( 84 \).

Thus, confirming the answer, Samanta can make 84 different selections of three toys.

Pritika is using the Binomial Theorem to find the binomial expansion of (a+b)7
. Which of the following is a term in the expansion?(1 point)
Responses

21a2b5
21 a squared b superscript 5 baseline

a4b3
a superscript 4 baseline b cubed

21a6b
21 a superscript 6 baseline b

21a2b4

To find a term in the expansion of \((a + b)^7\) using the Binomial Theorem, we can use the formula:

\[ \binom{n}{k} a^{n-k} b^k \]

where \(n\) is the exponent of the binomial (in this case, \(n = 7\)), and \(k\) varies from \(0\) to \(n\).

Step-by-Step Selection of Terms

Let's break down the terms given in the options and check which one can be generated from this expansion:

  1. Term: \(21a^2b^5\)

    • For this term, we identify \(n-k=2\) (power of \(a\)) and \(k=5\) (power of \(b\)).
    • So, \(n = k + (n-k) = 5 + 2 = 7\), which is valid.
    • The coefficient can be calculated as: \[ \binom{7}{5} = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]
    • Hence, this term is \(21a^2b^5\).

  2. Term: \(a^4b^3\)

    • Here, \(n-k=4\) and \(k=3\).
    • So, \(n = 4 + 3 = 7\), which is valid.
    • The coefficient: \[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]
    • Hence, this term is \(35a^4b^3\), not \(a^4b^3\).

  3. Term: \(21a^6b\)

    • Here, \(n-k=6\) and \(k=1\).
    • Thus, \(n = 6 + 1 = 7\), which is valid.
    • The coefficient: \[ \binom{7}{1} = 7 \]
    • Hence, this term is \(7a^6b\), not \(21a^6b\).

  4. Term: \(21a^2b^4\)

    • For this term, \(n-k=2\) and \(k=4\).
    • Therefore, \(n = 4 + 2 = 6\), which is not valid.

Conclusion

From the evaluation of the terms based on the Binomial Theorem, the valid term from the options given is:

21a²b⁵.

Thus, the correct answer is 21a²b⁵.

A cube is shrunk so that its side lengths of 2x
are reduced by 3 units. Using the Binomial Theorem, which of the following is the correct expression for the new volume of the cube?(1 point)
Responses

8x3−36x2+54x+27
8 x cubed minus 36 x squared plus 54 x plus 27

8x3+36x2+54x−27
8 x cubed plus 36 x squared plus 54 x minus 27

8x3−36x2+54x−27
8 x cubed minus 36 x squared plus 54 x minus 27

8x3+36x2+54x+27
8 x cubed plus 36 x squared plus 54 x plus 27

To find the new volume of the cube after its side lengths of \(2x\) are reduced by 3 units, we can follow these steps:

  1. Determine the original side length and new side length:

    • The original side length of the cube is \(2x\).
    • The new side length after reducing 3 units is \(2x - 3\).

  2. Calculate the volume of the cube using the new side length:

    • Volume \(V\) of a cube with side length \(s\) is given by: \[ V = s^3 \]
    • Therefore, the new volume is: \[ V_{\text{new}} = (2x - 3)^3 \]

  3. Expand \( (2x - 3)^3 \) using the Binomial Theorem:

    • The Binomial Theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
    • In this case, \(a = 2x\), \(b = -3\), and \(n = 3\).

    Thus, \[ (2x - 3)^3 = (2x)^3 + \binom{3}{1} (2x)^2 (-3) + \binom{3}{2} (2x)^1 (-3)^2 + \binom{3}{3} (-3)^3 \]

    Calculating each term:

    • First term: \((2x)^3 = 8x^3\)
    • Second term: \(\binom{3}{1} (2x)^2 (-3) = 3 \cdot 4x^2 \cdot (-3) = -36x^2\)
    • Third term: \(\binom{3}{2} (2x)^1 (-3)^2 = 3 \cdot 2x \cdot 9 = 54x\)
    • Fourth term: \(\binom{3}{3} (-3)^3 = -27\)

  4. Combine all the terms: \[ (2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27 \]

Thus, the new volume of the cube is given by the expression:

\[ \boxed{8x^3 - 36x^2 + 54x - 27} \]

The correct response is: 8 x cubed minus 36 x squared plus 54 x minus 27.