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

A. 3x^2-5x+4
B. 3x(x)+3x(-4)+-1(x)+-1(-4)
C. 3x^2+6x+4
D. 3x(-1)+3x(3x)+x(-4)+x(x)

9 answers

To rewrite the algebraic expression \((3x-1)(x-4)\) using the Distributive Property, let's apply the FOIL method (First, Outside, Inside, Last):

  1. First: Multiply the first terms: \(3x \cdot x = 3x^2\)
  2. Outside: Multiply the outer terms: \(3x \cdot (-4) = -12x\)
  3. Inside: Multiply the inner terms: \(-1 \cdot x = -1x\)
  4. Last: Multiply the last terms: \(-1 \cdot (-4) = 4\)

Next, combine all the results:

\[ 3x^2 - 12x - 1x + 4 = 3x^2 - 13x + 4 \]

Now, let's evaluate the provided options:

A. \(3x^2 - 5x + 4\) (Incorrect) B. \(3x(x) + 3x(-4) + -1(x) + -1(-4)\) (This is the correct expansion form) C. \(3x^2 + 6x + 4\) (Incorrect) D. \(3x(-1) + 3x(3x) + x(-4) + x(x)\) (Incorrect)

Thus, the correct way to rewrite the expression using the Distributive Property from the choices given is:

B. \(3x(x) + 3x(-4) + -1(x) + -1(-4)\)

Use the Distributive Property to find which expression is equivalent to 9x^2-25.
A. (4.5x+12.5)(4.5x-12.5)
B. 9x(x-25)
C. (3x+5)(3x-5)
D. (3x+5)^2

To find which expression is equivalent to \(9x^2 - 25\), we can recognize that this expression is a difference of squares. The general form for the difference of squares is:

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

In this case, we can rewrite \(9x^2 - 25\) as:

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

Now, applying the difference of squares formula:

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

Let's check the answer choices:

A. \((4.5x + 12.5)(4.5x - 12.5)\) (Not correct; not a difference of squares matching the original expression) B. \(9x(x - 25)\) (This does not represent the original expression) C. \((3x + 5)(3x - 5)\) (This is correct; it matches our factored form) D. \((3x + 5)^2\) (This represents a square, not a difference of squares)

The correct answer is:

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

Use the Distributive Property to verify which expression is equivalent to (x-5)(x-3).
A. x^2-8x+15
B. 2x-2
C. 3x^2-4x^2-3x-15
D. x^2-2x + 15

To verify which expression is equivalent to \((x - 5)(x - 3)\) using the Distributive Property (or FOIL method), we'll expand the expression:

Using FOIL:

  1. First: \(x \cdot x = x^2\)
  2. Outside: \(x \cdot (-3) = -3x\)
  3. Inside: \(-5 \cdot x = -5x\)
  4. Last: \(-5 \cdot (-3) = 15\)

Now, combine all of these results:

\[ x^2 - 3x - 5x + 15 = x^2 - 8x + 15 \]

Now let's check the answer choices:

A. \(x^2 - 8x + 15\) (This is correct; it matches our expansion) B. \(2x - 2\) (This is not correct) C. \(3x^2 - 4x^2 - 3x - 15\) (This simplifies to \(-x^2 - 3x - 15\), which is not correct) D. \(x^2 - 2x + 15\) (This is not correct)

The correct answer is:

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

Which expression is equivalent to (z+4)(z-4)?
A. 2z^2 - z^2 - 8
B. z^2 + 4z - 4z -16
C. z(z) + 4z(z) - 4(4)
D. z^2 - 8z - 16

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

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

In this case, we have:

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

Now, let's evaluate the given options:

A. \(2z^2 - z^2 - 8\)

  • This simplifies to \(z^2 - 8\) (Not correct)

B. \(z^2 + 4z - 4z - 16\)

  • This simplifies to \(z^2 - 16\) (Correct)

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

  • This would not simplify correctly; it mixes terms incorrectly. We should interpret the expression carefully but it does not match.

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

  • This expression does not match; it includes an extra \(-8z\).

The correct answer is:

B. \(z^2 - 16\) (after simplification).

What is another way to write the expression 2(x+1)(x+2)?
A. 2x^2+3x+2
B. 2x^2+6x+4
C. 2x^2+5x+2
D. x^2+6x+2

To rewrite the expression \(2(x + 1)(x + 2)\), we can first expand the expression inside the parentheses and then distribute \(2\).

  1. Expand \((x + 1)(x + 2)\) using the distributive property (or FOIL):

    • First: \(x \cdot x = x^2\)
    • Outside: \(x \cdot 2 = 2x\)
    • Inside: \(1 \cdot x = 1x\)
    • Last: \(1 \cdot 2 = 2\)

    Combining these results gives:

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

  2. Now, multiply the entire result by \(2\):

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

Now, let's compare this final expression \(2x^2 + 6x + 4\) against the options given:

A. \(2x^2 + 3x + 2\) (Not correct)

B. \(2x^2 + 6x + 4\) (This is correct)

C. \(2x^2 + 5x + 2\) (Not correct)

D. \(x^2 + 6x + 2\) (Not correct)

Thus, the correct answer is:

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