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multiply the polynomial expression (x+2)(x-4)

__x^2+__x+__

5 answers

To multiply the polynomial expressions \((x+2)(x-4)\), you can use the distributive property (also known as the FOIL method for binomials).

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

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

  3. Multiply the inner terms: \[ 2 \cdot x = 2x \]

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

Now combine all these results: \[ x^2 - 4x + 2x - 8 \]

Combine the like terms: \[ x^2 - 2x - 8 \]

Thus, the final result is: \[ x^2 - 2x - 8 \]

In the blanks, you would fill it as: __1 for \(x^2\), __-2 for \(x\), and __-8 for the constant term.

So the result is:

  • First blank: 1
  • Second blank: -2
  • Third blank: -8

Final answer: \(x^2 - 2x - 8\)

multiply to simplify the polynomial expression (r-1)(r^2-2r+3)
__r^3+__r^2+__r+__

To multiply and simplify the polynomial expression \((r-1)(r^2-2r+3)\), we will distribute each term in the first polynomial to every term in the second polynomial.

  1. Multiply \(r\) by each term in \(r^2 - 2r + 3\):

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

  2. Multiply \(-1\) by each term in \(r^2 - 2r + 3\):

    • \(-1 \cdot r^2 = -r^2\)
    • \(-1 \cdot (-2r) = 2r\)
    • \(-1 \cdot 3 = -3\)

Now, let's combine all the results together: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Next, we will combine the like terms:

  • For \(r^2\): \(-2r^2 - r^2 = -3r^2\)
  • For \(r\): \(3r + 2r = 5r\)

So, the expression simplifies to: \[ r^3 - 3r^2 + 5r - 3 \]

Finally, filling the given blanks:

  • First blank (for \(r^3\)): 1
  • Second blank (for \(r^2\)): -3
  • Third blank (for \(r\)): 5
  • Fourth blank (for the constant): -3

Thus, the completed expression is: \[ \mathbf{1}r^3 + \mathbf{-3}r^2 + \mathbf{5}r + \mathbf{-3} \]

simplify the polynomial expression (xy-2)(x^2+1)
__x^3y+__x^2+__xy+__

To simplify the polynomial expression \((xy - 2)(x^2 + 1)\), we will distribute each term in the first polynomial to each term in the second polynomial.

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

    • \(xy \cdot x^2 = x^3y\)
    • \(xy \cdot 1 = xy\)

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

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

Now we combine all the results together: \[ x^3y + xy - 2x^2 - 2 \]

Next, we can arrange the terms based on the powers of \(x\):

  • \(x^3y\)
  • \(-2x^2\)
  • \(xy\)
  • \(-2\)

Thus, the expression simplifies to: \[ x^3y - 2x^2 + xy - 2 \]

Now, filling in the blanks:

  • First blank (for \(x^3y\)): 1
  • Second blank (for \(x^2\)): -2
  • Third blank (for \(xy\)): 1
  • Fourth blank (for the constant): -2

So, the final answer is: \[ \mathbf{1}x^3y + \mathbf{-2}x^2 + \mathbf{1}xy + \mathbf{-2} \]