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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

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

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

3x2−5x+4

9 answers

To rewrite the algebraic expression \((3x - 1)(x - 4)\) using the Distributive Property (also known as the FOIL method for binomials), we multiply each term in the first binomial by each term in the second binomial.

Let's apply the distributive property:

  1. First terms: \(3x \cdot x = 3x^2\)
  2. Outer terms: \(3x \cdot (-4) = -12x\)
  3. Inner terms: \(-1 \cdot x = -1x\)
  4. Last terms: \(-1 \cdot (-4) = 4\)

Now, combining these results: \[ 3x^2 - 12x - 1x + 4 \] Combine the like terms \(-12x - 1x\): \[ 3x^2 - 13x + 4 \]

Thus, the correct answer is not provided in your options. However, the fully simplified expression is:

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

If you are looking for the best match from the provided options, the first response appears to be completely incorrect, and the option involving "3x2−5x+4" is inaccurate as well. The expression needs to be correctly calculated to reveal \(3x^2 - 13x + 4\).

For more clarity: None of the provided responses are completely correct, and it's important to calculate it as shown to get the final expression.

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

(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
left parenthesis 3 x plus 5 right parenthesis squared

(3x+5)(3x−5)

To determine which expression is equivalent to \(9x^2 - 25\), we can recognize that this expression can be factored using the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

In our case:

  • \(9x^2\) is a perfect square and can be written as \((3x)^2\).
  • \(25\) is also a perfect square and can be written as \(5^2\).

So, we can apply the difference of squares:

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

Now, let’s evaluate the given options:

  1. \(9x(x - 25)\): Not equivalent.
  2. \((4.5x + 12.5)(4.5x - 12.5)\): Not equivalent.
  3. \((3x + 5)^2\): Not equivalent.
  4. \((3x + 5)(3x - 5)\): This is equivalent as shown in our factorization.

Thus, the equivalent expression to \(9x^2 - 25\) is:

\[ (3x + 5)(3x - 5) \]

So the correct response is:

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

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

x2−2x+15
x squared minus 2 x plus 15

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

2x−2
2 x minus 2

x2−8x+15

To verify which expression is equivalent to \((x - 5)(x - 3)\), we can use the Distributive Property (also known as the FOIL method for binomials). Here’s how it works:

  1. First terms: \(x \cdot x = x^2\)
  2. Outer terms: \(x \cdot (-3) = -3x\)
  3. Inner terms: \(-5 \cdot x = -5x\)
  4. Last terms: \(-5 \cdot (-3) = 15\)

Now, we combine these results:

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

Combine the like terms (\(-3x\) and \(-5x\)):

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

So, \((x - 5)(x - 3) = x^2 - 8x + 15\).

Now, let's evaluate the given responses:

  1. \(x^2 - 2x + 15\): Not equivalent.
  2. \(3x^2 - 4x^2 - 3x - 15\): This expression simplifies to \(-x^2 - 3x - 15\), which is not equivalent.
  3. \(2x - 2\): Not equivalent.
  4. \(x^2 - 8x + 15\): This is the equivalent expression we derived from \((x - 5)(x - 3)\).

The correct response 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

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

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

z2−8z−16

To determine which expression is equivalent to \((z + 4)(z - 4)\), we can recognize that this expression can be factored using the difference of squares. The formula for the difference of squares is:

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

In this case:

  • \(a = z\)
  • \(b = 4\)

So, we apply the difference of squares:

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

Now, let's evaluate the given responses:

  1. \(z^2 + 4z - 4z - 16\): This simplifies to \(z^2 - 16\), which is equivalent.
  2. \(2z^2 - z^2 - 8\): This simplifies to \(z^2 - 8\), which is not equivalent.
  3. \(z(z) + 4z(z) - 4(4)\): This simplifies to \(z^2 + 4z^2 - 16 = 5z^2 - 16\), which is not equivalent.
  4. \(z^2 - 8z - 16\): This is not equivalent.

The correct response that is 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

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

2x2+6x+4
2 x squared plus 6 x plus 4

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

2x2+5x+2

To rewrite the expression \(2(x + 1)(x + 2)\), we need to first expand the expression inside the parentheses using the distributive property (or the FOIL method) and then multiply by 2.

  1. First, expand \((x + 1)(x + 2)\):

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

  2. Now, multiply the result by 2:

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

Now we can evaluate the given responses:

  1. \(x^2 + 6x + 2\): This is not equivalent.
  2. \(2x^2 + 6x + 4\): This is equivalent to our expansion.
  3. \(2x^2 + 3x + 2\): This is not equivalent.
  4. \(2x^2 + 5x + 2\): This is not equivalent.

The expression that correctly represents \(2(x + 1)(x + 2)\) is:

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