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According to the Distributive Property, which is a way to rewrite the algebraic expression (3x−1)(x−4) ?(1 point)

Responses

3x(x)+3x(−4)+−1(x)+−1(−4)
3 x left parenthesis x right parenthesis plus 3 x left parenthesis negative 4 right parenthesis plus negative 1 left parenthesis x right parenthesis plus negative 1 left parenthesis negative 4 right parenthesis

3x2+6x+4
3 x squared plus 6 x plus 4

3x(−1)+3x(3x)+x(−4)+x(x)
3 x left parenthesis negative 1 right parenthesis plus 3 x left parenthesis 3 x right parenthesis plus x left parenthesis negative 4 right parenthesis plus x left parenthesis x right parenthesis

3x2−5x+4

9 answers

To apply the Distributive Property to the expression \( (3x - 1)(x - 4) \), we distribute each term in the first parentheses by each term in the second parentheses:

  1. \( 3x \times x = 3x^2 \)
  2. \( 3x \times (-4) = -12x \)
  3. \( -1 \times x = -x \)
  4. \( -1 \times (-4) = 4 \)

Now, combine the results:

\[ 3x^2 - 12x - x + 4 \]

Combine the like terms \( -12x - x = -13x \):

So, the expression simplifies to:

\[ 3x^2 - 13x + 4 \]

From the given options, none of the options are correct as the one that correctly represents the distributive property and simplifies is not among them. However, the option that resembles the distributive property application most closely is:

3x(x) + 3x(-4) + -1(x) + -1(-4)

This option correctly shows the application of the distributive property, but it simplifies further to \( 3x^2 - 13x + 4\).

Use the Distributive Property to find which expression is equivalent to 9x2−25 .(1 point)
Responses

9x(x−25)
9 x left parenthesis x minus 25 right parenthesis

(3x+5)(3x−5)
left parenthesis 3 x plus 5 right parenthesis left parenthesis 3 x minus 5 right parenthesis

(4.5x+12.5)(4.5x−12.5)
left parenthesis 4.5 x plus 12.5 right parenthesis left parenthesis 4.5 x minus 12.5 right parenthesis

(3x+5)2

The expression \( 9x^2 - 25 \) can be recognized as a difference of squares, which has the general form \( a^2 - b^2 = (a - b)(a + b) \).

In this case:

  • \( a^2 = 9x^2 \) implies \( a = 3x \)
  • \( b^2 = 25 \) implies \( b = 5 \)

Thus, we can rewrite \( 9x^2 - 25 \) as:

\[ (3x)^2 - (5)^2 = (3x - 5)(3x + 5) \]

Now, let's check the given options to see which matches:

  1. \( 9x(x - 25) \) - This is not correct.
  2. \( (3x + 5)(3x - 5) \) - This matches our result.
  3. \( (4.5x + 12.5)(4.5x - 12.5) \) - This is not correct.
  4. \( (3x + 5)^2 \) - This is not correct as it's a square, not a difference of squares.

The correct expression equivalent to \( 9x^2 - 25 \) is:

\( (3x + 5)(3x - 5) \).

Use the Distributive Property to verify which expression is equivalent to (x−5)(x−3) .(1 point)
Responses

x2−8x+15
x squared minus 8 x plus 15

3x2−4x2−3x−15
3 x squared minus 4 x squared minus 3 x minus 15

x2−2x+15
x squared minus 2 x plus 15

2x−2

To verify which expression is equivalent to \( (x - 5)(x - 3) \) using the Distributive Property, we can expand the product:

  1. Distribute \( x \) across \( (x - 3) \):

    • \( x \times x = x^2 \)
    • \( x \times (-3) = -3x \)

  2. Distribute \( -5 \) across \( (x - 3) \):

    • \( -5 \times x = -5x \)
    • \( -5 \times (-3) = 15 \)

Now combine all the terms:

\[ x^2 - 3x - 5x + 15 \]

This simplifies to:

\[ x^2 - 8x + 15 \]

Now let's check the provided options for equivalence:

  1. \( x^2 - 8x + 15 \) - This matches our result.
  2. \( 3x^2 - 4x^2 - 3x - 15 \) - This does not simplify to a usable form matching our result.
  3. \( x^2 - 2x + 15 \) - This does not match our result.
  4. \( 2x - 2 \) - This does not match our result.

The correct expression equivalent to \( (x - 5)(x - 3) \) is:

\( x^2 - 8x + 15 \).

Which expression is equivalent to (z+4)(z−4) ?(1 point)
Responses

z2+4z−4z−16
z squared plus 4 z minus 4 z minus 16

z2−8z−16
z squared minus 8 z minus 16

2z2−z2−8
2 z squared minus z squared minus 8

z(z)+4z(z)−4(4)

To find the expression equivalent to \( (z+4)(z-4) \), we can recognize that this is a difference of squares, which has the general form \( a^2 - b^2 \):

\[ (z + 4)(z - 4) = z^2 - 4^2 = z^2 - 16 \]

Now, let's review the options provided to see which one matches:

  1. \( z^2 + 4z - 4z - 16 \):

    • This simplifies to \( z^2 - 16 \) (since \( 4z - 4z = 0\)), which is correct.

  2. \( z^2 - 8z - 16 \):

    • This does not match \( z^2 - 16 \).

  3. \( 2z^2 - z^2 - 8 \):

    • This simplifies to \( z^2 - 8 \), which does not match.

  4. \( z(z) + 4z(z) - 4(4) \):

    • This expands to \( z^2 + 4z^2 - 16 = 5z^2 - 16 \), which does not match.

The correct expression equivalent to \( (z+4)(z-4) \) is:

\( z^2 + 4z - 4z - 16 \), which simplifies to \( z^2 - 16 \).

What is another way to write the expression 2(x+1)(x+2) ?(1 point)
Responses

2x2+5x+2
2 x squared plus 5 x plus 2

x2+6x+2
x squared plus 6 x plus 2

2x2+3x+2
2 x squared plus 3 x plus 2

2x2+6x+4

To rewrite the expression \( 2(x + 1)(x + 2) \), we can first expand the product \( (x + 1)(x + 2) \) using the distributive property (also known as the FOIL method):

  1. Use FOIL:
    • First: \( x \times x = x^2 \)
    • Outside: \( x \times 2 = 2x \)
    • Inside: \( 1 \times x = x \)
    • Last: \( 1 \times 2 = 2 \)

Now, combine these results:

\[ (x + 1)(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2 \]

Now, include the factor of 2:

\[ 2(x^2 + 3x + 2) = 2x^2 + 6x + 4 \]

Now let's check the provided options to see which one matches:

  1. \( 2x^2 + 5x + 2 \) - This does not match.
  2. \( x^2 + 6x + 2 \) - This does not match.
  3. \( 2x^2 + 3x + 2 \) - This does not match.
  4. \( 2x^2 + 6x + 4 \) - This matches our result.

Therefore, another way to write the expression \( 2(x + 1)(x + 2) \) is:

\( 2x^2 + 6x + 4 \).