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use the Distributive property to rewrite the polynomial (x+8)(x-2)

__z^2+__x+__

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To use the distributive property (also known as the FOIL method for binomials), we need to multiply each term in the first binomial by each term in the second binomial.

Given the expression \((x + 8)(x - 2)\):

  1. First: Multiply the first terms:
    \(x \cdot x = x^2\)

  2. Outer: Multiply the outer terms:
    \(x \cdot (-2) = -2x\)

  3. Inner: Multiply the inner terms:
    \(8 \cdot x = 8x\)

  4. Last: Multiply the last terms:
    \(8 \cdot (-2) = -16\)

Now, let's combine these results:

\[ x^2 - 2x + 8x - 16 \]

Combine like terms (-2x and 8x):

\[ x^2 + 6x - 16 \]

So, the polynomial \((x + 8)(x - 2)\) can be rewritten as:

\[ x^2 + 6x - 16 \]

Therefore, the completed blanks are:

  • __z^2 = 1 (coefficient of \(x^2\))
  • __x = 6 (coefficient of \(x\))
  • __ = -16 (constant term)

Thus, the final result is: 1 for \(z^2\), 6 for \(x\), and -16 for the constant.

rewrite the expression 3x(x+3) using the Distributive property

To rewrite the expression \(3x(x + 3)\) using the Distributive Property, you will multiply \(3x\) by each term inside the parentheses.

Here are the steps:

  1. Distribute \(3x\) to \(x\): \[ 3x \cdot x = 3x^2 \]

  2. Distribute \(3x\) to \(3\): \[ 3x \cdot 3 = 9x \]

Now, combine the results. The expression \(3x(x + 3)\) can be rewritten as: \[ 3x^2 + 9x \]

So, \(3x(x + 3) = 3x^2 + 9x\).

Find the product of the polynomials (2x+1)(x-4)
__x^2+__x+__

To find the product of the polynomials \((2x + 1)(x - 4)\), we can use the distributive property (or the FOIL method for binomials).

Steps:

  1. First: Multiply the first terms: \[ 2x \cdot x = 2x^2 \]

  2. Outer: Multiply the outer terms: \[ 2x \cdot (-4) = -8x \]

  3. Inner: Multiply the inner terms: \[ 1 \cdot x = 1x \quad (\text{or just } x) \]

  4. Last: Multiply the last terms: \[ 1 \cdot (-4) = -4 \]

Combine the results:

Now, sum all the terms obtained: \[ 2x^2 - 8x + 1x - 4 \]

Combine like terms (\(-8x + 1x\)): \[ 2x^2 - 7x - 4 \]

So, the product of the polynomials \((2x + 1)(x - 4)\) can be rewritten as: \[ 2x^2 - 7x - 4 \]

Therefore, filling in the blanks:

  • __x^2 = 2
  • __x = -7
  • __ = -4

The final result is: 2 for \(x^2\), -7 for \(x\), and -4 for the constant.

according the the distributive property which is the right way to rewrite the algebraic expression (3x-1)(x-4)?
A. 3x^2+6x+4
B. 3x(x)+3x(-4)+-1(x)+-1(-4)
C. 3x^2-5+4
D. 3x(-1)+3x(3x)+x(-4)+x(x)

To correctly apply the distributive property to the expression \((3x - 1)(x - 4)\), let's clarify how the distributive property works step by step.

Using the distributive property (FOIL method in the case of binomials), we multiply each term in the first polynomial by each term in the second polynomial:

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

Combining these results gives: \[ 3x^2 - 12x - 1x + 4 = 3x^2 - 13x + 4 \]

Now, let's assess the given options:

A. \(3x^2 + 6x + 4\)
(Not correct; wrong coefficients and terms.)

B. \(3x(x) + 3x(-4) + -1(x) + -1(-4)\)
(This is correct; it accurately represents the terms from the distribution.)

C. \(3x^2 - 5 + 4\)
(Not correct; this is incorrect due to wrong simplification of terms.)

D. \(3x(-1) + 3x(3x) + x(-4) + x(x)\)
(Not correct; misrepresents the terms.)

Conclusion:

The correct way to rewrite the algebraic expression \((3x - 1)(x - 4)\) according to the distributive property 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. (3x+5)(3x-5)
B. (3x+5)^2
C. 9x(x-25)
D. (4.5x+12.5)(4.5x-12.5)

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

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

In our case, we have:

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

Using the formula, we can factor \(9x^2 - 25\):

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

Now let's evaluate the options provided:

A. \((3x + 5)(3x - 5)\)
(This is correct; it matches our factorization.)

B. \((3x + 5)^2\)
(Not correct; this would be \(9x^2 + 30x + 25\).)

C. \(9x(x - 25)\)
(Not correct; this doesn't represent the original expression.)

D. \((4.5x + 12.5)(4.5x - 12.5)\)
(Not correct; this would represent a different expression.)

Conclusion:

The expression that is equivalent to \(9x^2 - 25\) is A: \[ (3x + 5)(3x - 5) \]

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

To determine which expression is equivalent to \((x - 5)(x - 3)\) using the distributive property, we can expand the expression step by step:

Using the Distributive Property (FOIL Method):

  1. First: Multiply the first terms: \[ x \cdot x = x^2 \]

  2. Outer: Multiply the outer terms: \[ x \cdot (-3) = -3x \]

  3. Inner: Multiply the inner terms: \[ -5 \cdot x = -5x \]

  4. Last: Multiply the last terms: \[ -5 \cdot (-3) = 15 \]

Combine these results:

Now, we add all the terms together: \[ x^2 - 3x - 5x + 15 \]

Combine like terms (\(-3x - 5x\)): \[ x^2 - 8x + 15 \]

Conclusion:

The expression \((x - 5)(x - 3)\) simplifies to: \[ x^2 - 8x + 15 \]

Now, let's evaluate the options:

A. \(x^2 - 2x + 15\)
(Not correct; incorrect coefficient for \(x\).)

B. \(x^2 - 8x + 15\)
(This is correct; it matches our expansion.)

C. \(2x - 2\)
(Not correct; this does not represent the original expression.)

D. \(3x^2 - 4x^2 - 3x - 15\)
(Not correct; this simplifies to \(-x^2 - 3x - 15\), which also does not match.)

The correct answer is B: \[ x^2 - 8x + 15 \]

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

To find an equivalent expression to \((z + 4)(z - 4)\), we can recognize that this is a difference of squares. The difference of squares formula states that:

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

In this case:

  • \(a = z\)
  • \(b = 4\)

Thus, we can express it as:

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

Now, let's evaluate the options provided to see which one is equivalent to \(z^2 - 16\):

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

  • This simplifies to: \[ 2z^2 - z^2 - 8 = z^2 - 8 \quad \text{(Not correct)} \]

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

  • This does not match because it has both a linear term and the wrong constant term. \quad \text{(Not correct)}

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

  • This simplifies to: \[ z^2 + 4z - 4z - 16 = z^2 - 16 \quad \text{(This is correct!)} \]

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

  • This expression does not simplify to the desired form. In fact, it is convoluted and doesn't match our expression. \quad \text{(Not correct)}

Conclusion:

The expression equivalent to \((z + 4)(z - 4)\) is C: \[ z^2 + 4z - 4z - 16 = z^2 - 16 \]