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Multiply the polynomial expression (x+2)(x−4)

(needs number)x^2 + (needs number)x +(needs number)

11 answers

To multiply the polynomial expression \((x + 2)(x - 4)\), we can use the distributive property (also known as the FOIL method for binomials). Here are the steps:

  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 \(-4x\) and \(2x\): \[ x^2 - 2x - 8 \]

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

So, the expression can be written as: \[ 1x^2 + (-2)x + (-8) \]

In the format you requested:

  • The coefficient of \(x^2\) is 1.
  • The coefficient of \(x\) is -2.
  • The constant term is -8.
Multiply to simplify the polynomial expression (r−1)(r2−2r+3)
.(2 points)
r3+
r2+
r+

To simplify the expression \((r - 1)(r^2 - 2r + 3)\), we will use the distributive property, multiplying each term in the first polynomial by each term in the second polynomial.

Here are the steps:

  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, combine all these results: \[ r^3 - 2r^2 + 3r - r^2 + 2r - 3 \]

Next, combine the like terms:

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

Putting it all together, we have: \[ r^3 - 3r^2 + 5r - 3 \]

Therefore, the final simplified expression can be written as: \[ r^3 + (-3)r^2 + 5r + (-3) \]

In the format you requested:

  • The coefficient of \(r^3\) is 1.
  • The coefficient of \(r^2\) is -3.
  • The coefficient of \(r\) is 5.
  • The constant term is -3.
Simplify the polynomial expression (xy−2)(x2+1)
.(2 points)
x3y+
x2+
xy+

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

Here are the steps:

  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, combine all these results: \[ x^3y + xy - 2x^2 - 2 \]

This can be arranged in standard form: \[ x^3y - 2x^2 + xy - 2 \]

Thus, in the format you requested:

  • The coefficient of \(x^3y\) is 1.
  • The coefficient of \(x^2\) is -2.
  • The coefficient of \(xy\) is 1.
  • The constant term is -2.

So the simplified expression can be written as: \[ x^3y + (-2)x^2 + xy + (-2) \]

Which of the following shows that polynomials form a closed system under multiplication?

Option #1: 3(2x+5)=6x+15


Option #2: 8x+7=5x+3x+1+6


Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24


Option #4: 2x12⋅3=6x−−√

To determine which option shows that polynomials form a closed system under multiplication, we need to identify which of the given expressions demonstrates that the product of two polynomials is also a polynomial.

  1. Option #1: \(3(2x + 5) = 6x + 15\)

    • This expression represents the multiplication of a polynomial \(2x + 5\) by a constant \(3\). The result is still a polynomial. However, this does not show multiplication between two polynomials.

  2. Option #2: \(8x + 7 = 5x + 3x + 1 + 6\)

    • This expression demonstrates how to combine like terms but does not involve multiplication at all. So, it does not show closure under multiplication.

  3. Option #3: \((2x^2 - 4)(3y + 6) = 6x^2y + 12x^2 - 12y - 24\)

    • This expression involves the multiplication of two polynomials: \(2x^2 - 4\) and \(3y + 6\). The result is also a polynomial, specifically \((2x^2 - 4)(3y + 6)\), confirming that polynomials are closed under multiplication.

  4. Option #4: \(2x^{12} \cdot 3 = 6x\sqrt{}\)

    • This option appears to have formatting issues, but the term \(2x^{12} \cdot 3 = 6x\sqrt{}\) does not show proper polynomial multiplication and likely does not yield a polynomial.

Given this analysis, Option #3 is the correct option that demonstrates that polynomials form a closed system under multiplication.

Multiply the polynomial (b+8)(3b−6) to simplify.(1 point)
Responses

3b2+18b−48
3 b squared plus 18 b minus 48

3b2+30b−48
3 b squared plus 30 b minus 48

b2+30b−48
b squared plus 30 b minus 48

b2+18b−36

To simplify the expression \((b + 8)(3b - 6)\), we will use the distributive property (FOIL method):

  1. Multiply the first terms:
    \[ b \cdot 3b = 3b^2 \]

  2. Multiply the outer terms:
    \[ b \cdot (-6) = -6b \]

  3. Multiply the inner terms:
    \[ 8 \cdot 3b = 24b \]

  4. Multiply the last terms:
    \[ 8 \cdot (-6) = -48 \]

Now, combine all these results together: \[ 3b^2 - 6b + 24b - 48 \]

Next, combine the like terms: \[ -6b + 24b = 18b \]

Putting it all together, we get: \[ 3b^2 + 18b - 48 \]

Thus, the simplified expression is: \[ 3b^2 + 18b - 48 \]

From the given responses, the correct answer is: 3b² + 18b - 48.

What is the product of the polynomials (x2y+2)(x2−y)?(1 point)
Responses

x3y−x2y2+2x2−2y
x cubed y minus x squared y squared plus 2 x squared minus 2 y

x4y−xy+x2y
x superscript 4 baseline y minus x y plus x squared y

x4y−x2y2+2x2−2y
x superscript 4 baseline y minus x squared y squared plus 2 x squared minus 2 y

x4y+x2−2y

To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will use the distributive property (FOIL method):

  1. Multiply the first terms: \[ x^2y \cdot x^2 = x^4y \]

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

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

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

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

Thus, the final product of the polynomials \((x^2y + 2)(x^2 - y)\) is: \[ x^4y - x^2y^2 + 2x^2 - 2y \]

From the given responses, the correct answer is: \(x^4y - x^2y^2 + 2x^2 - 2y\).