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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)
3x2−5x+4
3x(−1)+3x(3x)+x(−4)+x(x)
3x2+6x+4

9 answers

To apply the Distributive Property to the expression \((3x - 1)(x - 4)\), we can expand it as follows:

  1. Multiply \(3x\) by each term in the second parenthesis:

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

  2. Multiply \(-1\) by each term in the second parenthesis:

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

Now, combine all the results: \[ 3x^2 - 12x - x + 4 \]

Combine like terms: \[ 3x^2 - 13x + 4 \]

Now, let's review the options provided:

  1. 3x(x) + 3x(−4) + −1(x) + −1(−4) (Not correct, it doesn't simplify properly.)
  2. 3x^2 − 5x + 4 (Not correct, incorrect coefficient for x.)
  3. 3x(−1) + 3x(3x) + x(−4) + x(x) (Not correct, doesn't represent proper distribution.)
  4. 3x^2 + 6x + 4 (Incorrect due to wrong coefficients.)

None of the given responses correspond correctly to the expanded form \(3x^2 - 13x + 4\).

Make sure to double-check the options, but based on proper distribution, none of the options provided are correct.

Use the Distributive Property to find which expression is equivalent to 9x2−25 .(1 point)
Responses:
9x(x−25).
(3x+5)2.
(4.5x+12.5)(4.5x−12.5).
(3x+5)(3x−5).

To factor the expression \(9x^2 - 25\), we can recognize that it is a difference of squares. The difference of squares can be written in the form:

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

In this case, we can identify \(a\) and \(b\) as follows:

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

Now, applying the difference of squares formula gives us:

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

Now, let’s review the options provided:

  1. 9x(x−25) (Not correct, does not represent the factoring correctly.)
  2. (3x + 5)² (Not correct, this represents a perfect square, not a difference of squares.)
  3. (4.5x + 12.5)(4.5x - 12.5) (Not correct, this is an unrelated product.)
  4. (3x + 5)(3x - 5) (This is correct; it matches our derived expression.)

Therefore, 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:
3x2−4x2−3x−15
x2−2x+15
x2−8x+15
2x−2

To verify which expression is equivalent to \((x - 5)(x - 3)\), let's use the Distributive Property to expand the expression.

Applying the Distributive Property:

  1. Multiply \(x\) by each term in the second parenthesis \((x - 3)\):

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

  2. Multiply \(-5\) by each term in the second parenthesis \((x - 3)\):

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

Now, combine all the results:

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

Combine like terms:

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

Now, let's review the options provided:

  1. 3x² - 4x² - 3x - 15 (Not correct, this does not simplify correctly.)
  2. x² - 2x + 15 (Not correct, it has an incorrect coefficient for \(x\).)
  3. x² - 8x + 15 (This matches our expansion exactly and is correct.)
  4. 2x - 2 (Not correct, this is not related to the expression.)

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

x² - 8x + 15.

Which expression is equivalent to (z+4)(z−4)?(1 point)
Responses:
2z2−z2−8
z(z)+4z(z)−4(4)
z2−8z−16
z2+4z−4z−16

To find the expression equivalent to \((z + 4)(z - 4)\), we can recognize that this expression represents a difference of squares, which is given by the formula:

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

For our expression:

  • \(a = z\)
  • \(b = 4\)

Using the difference of squares formula, we have:

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

Now let's review the options provided:

  1. 2z² - z² - 8 (This simplifies to \(z² - 8\), which is not correct.)
  2. z(z) + 4z(z) - 4(4) (This expression does not simplify correctly and is not in the appropriate form.)
  3. z² - 8z - 16 (This expression is incorrect; it has an incorrect linear term.)
  4. z² + 4z - 4z - 16 (This simplifies to \(z² - 16\), which is correct.)

So the expression equivalent to \((z + 4)(z - 4)\) is:

z² - 16 (which corresponds to option 4: z² + 4z - 4z - 16).

What is another way to write the expression 2(x+1)(x+2) ?
Responses:
2x2+6x+4
2x2+3x+2
2x2+5x+2
x2+6x+2

To rewrite the expression \(2(x + 1)(x + 2)\), we can first expand the expression \((x + 1)(x + 2)\) using the Distributive Property (also known as the FOIL method for binomials):

  1. Multiply \(x\) by each term in \((x + 2)\):

    • \(x \cdot x = x^2\)
    • \(x \cdot 2 = 2x\)

  2. Multiply \(1\) by each term in \((x + 2)\):

    • \(1 \cdot x = x\)
    • \(1 \cdot 2 = 2\)

Now, combine all the results from these multiplications:

\[ x^2 + 2x + x + 2 = x^2 + 3x + 2 \]

Next, we multiply this result by \(2\):

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

Now let's review the options you provided:

  1. 2x² + 6x + 4 (This matches our result perfectly.)
  2. 2x² + 3x + 2 (This does not match; it's just the result from \((x + 1)(x + 2)\) without multiplying by 2.)
  3. 2x² + 5x + 2 (This does not match.)
  4. x² + 6x + 2 (This does not match; it lacks the coefficient of 2 for \(x^2\).)

The correct expression equivalent to \(2(x + 1)(x + 2)\) is:

2x² + 6x + 4.