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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(−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

3x^2−5x+4
3 x squared minus 5 x plus 4

3x^2+6x+4
3 x squared plus 6 x plus 4

3x(x)+3x(−4)+−1(x)+−1(−4)

9 answers

To rewrite the algebraic expression \( (3x - 1)(x - 4) \) using the Distributive Property, we will need to distribute each term in the first parenthesis by each term in the second parenthesis.

Let's do the step-by-step distribution:

  1. Distribute \( 3x \):

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

  2. Distribute \( -1 \):

    • \( -1 \times x = -x \)
    • \( -1 \times (-4) = 4 \)

Now, we combine all of these results together:

  • \( 3x^2 - 12x - x + 4 \)

Next, combine the like terms:

  • The terms \( -12x \) and \( -x \) combine to \( -13x \).

So the final expression becomes:

  • \( 3x^2 - 13x + 4 \)

However, note that this expression does not match any options given, so let's check the expected responses closely:

  1. \( 3x(−1)+3x(3x)+x(−4)+x(x) \) - this is not correctly simplified.

  2. \( 3x^2−5x+4 \) - this does not match any calculations.

  3. \( 3x^2+6x+4 \) - this is incorrect according to our calculations.

  4. \( 3x(x)+3x(−4)+−1(x)+−1(−4) \) - this is a breakdown of the distribution, but it does not represent the simplified result.

Thus, it seems that none of these responses correctly represent the outcome of \( (3x - 1)(x - 4) \) after applying the Distributive Property correctly; however, option 4 shows the correct distribution steps before simplification.

Use the Distributive Property to find which expression is equivalent to 9x^2−25 .(1 point)
Responses

(3x+5)2
left parenthesis 3 x plus 5 right parenthesis squared

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

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

(4.5x+12.5)(4.5x−12.5)

To determine which expression is equivalent to \( 9x^2 - 25 \) using the Distributive Property, we can recognize that \( 9x^2 - 25 \) is a difference of squares, which can be factored using the formula:

\[ a^2 - b^2 = (a + b)(a - b) \]

In this case:

  • \( a^2 = 9x^2 \), which gives us \( a = 3x \)
  • \( b^2 = 25 \), which gives us \( b = 5 \)

Therefore, we can express \( 9x^2 - 25 \) as:

\[ 9x^2 - 25 = (3x)^2 - (5)^2 = (3x + 5)(3x - 5) \]

Now, let's analyze the provided responses:

  1. \( (3x + 5)^2 \): This expands to \( (3x + 5)(3x + 5) = 9x^2 + 30x + 25 \), which is not equivalent to \( 9x^2 - 25 \).

  2. \( (3x + 5)(3x - 5) \): This expands to \( 9x^2 - 25 \), which matches the original expression.

  3. \( 9x(x - 25) \): This expands to \( 9x^2 - 225x \), which is also not equivalent to \( 9x^2 - 25 \).

  4. \( (4.5x + 12.5)(4.5x - 12.5) \): This expands to a different expression. It calculates as \( (4.5x)^2 - (12.5)^2 = 20.25x^2 - 156.25 \), which does not match \( 9x^2 - 25 \).

Based on this analysis, the correct answer is:

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

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

x^2−8x+15
x squared minus 8 x plus 15

3x^2−4x^2−3x−15
3 x squared minus 4 x squared minus 3 x minus 15

x^2−2x+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 will expand the expression step by step.

Using the Distributive Property, we can expand \( (x - 5)(x - 3) \):

  1. Distribute \( x \):

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

  2. Distribute \( -5 \):

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

Now, combine all of these results: \[ x^2 - 3x - 5x + 15 \]

Next, combine the like terms \( -3x \) and \( -5x \): \[ x^2 - 8x + 15 \]

So, the expanded form of \( (x - 5)(x - 3) \) is: \[ x^2 - 8x + 15 \]

Now, let's check the provided responses:

  1. \( x^2 - 8x + 15 \) - This matches our calculated expansion.

  2. \( 3x^2 - 4x^2 - 3x - 15 \) - This simplifies to \( -x^2 - 3x - 15 \), which does not match.

  3. \( x^2 - 2x + 15 \) - This does not match our result.

  4. \( 2x - 2 \) - This does not match our result as well.

Therefore, the correct answer is:

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

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

z^2+4z−4z−16
z squared plus 4 z minus 4 z minus 16

z(z)+4z(z)−4(4)
z left parenthesis z right parenthesis plus 4 z left parenthesis z right parenthesis minus 4 left parenthesis 4 right parenthesis

z^2−8z−16
z squared minus 8 z minus 16

2z^2−z2−8

To find the expression equivalent to \( (z + 4)(z - 4) \), we can use the formula for the difference of squares:

\[ (a + b)(a - b) = a^2 - b^2 \]

In this case:

  • \( a = z \)
  • \( b = 4 \)

Applying the formula:

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

Now, let's evaluate the provided responses:

  1. \( z^2 + 4z - 4z - 16 \): This simplifies to \( z^2 - 16 \). (The \( +4z \) and \( -4z \) cancel out.)

  2. \( z(z) + 4z(z) - 4(4) \): This expands as \( z^2 + 4z^2 - 16 \). Combining like terms gives \( 5z^2 - 16 \), which is not correct.

  3. \( z^2 - 8z - 16 \): This is not correct since it introduces an unwanted \( -8z \) term.

  4. \( 2z^2 - z^2 - 8 \): This simplifies to \( z^2 - 8 \), which is also not correct.

Based on the calculations, the equivalent expression for \( (z + 4)(z - 4) \) is:

\( z^2 - 16 \), which corresponds correctly to the simplified form of the first response \( z^2 + 4z - 4z - 16 \) (as all terms confirmed).

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

2x^2+5x+2
2 x squared plus 5 x plus 2

x^2+6x+2
x squared plus 6 x plus 2

2x^2+3x+2
2 x squared plus 3 x plus 2

2x^2+6x+4

To find another way to write the expression \( 2(x + 1)(x + 2) \), we will first expand the expression inside the parentheses and then multiply by 2.

  1. Begin by expanding \( (x + 1)(x + 2) \) using the distributive property (or FOIL method):

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

  1. Now, multiply the entire expression by 2:

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

Now, let's analyze the provided responses:

  1. \( 2x^2 + 5x + 2 \): This does not match our result.

  2. \( x^2 + 6x + 2 \): This does not match our result.

  3. \( 2x^2 + 3x + 2 \): This does not match our result.

  4. \( 2x^2 + 6x + 4 \): This matches our result.

Therefore, the correct answer is:

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