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

b^2+30b−48
3b^2+18b−48
b^2+18b−36
3b^2+30b−48

11 answers

To multiply the polynomial \((b + 8)(3b - 6)\), use the distributive property (also known as the FOIL method for binomials):

\[ (b + 8)(3b - 6) = b(3b) + b(-6) + 8(3b) + 8(-6) \]

Calculating each term gives:

\[ b(3b) = 3b^2 \] \[ b(-6) = -6b \] \[ 8(3b) = 24b \] \[ 8(-6) = -48 \]

Now, combine all these terms:

\[ 3b^2 - 6b + 24b - 48 \]

Combine the like terms:

\[ 3b^2 + (−6b + 24b) - 48 = 3b^2 + 18b - 48 \]

Therefore, the simplified form of \((b + 8)(3b - 6)\) is:

\[ \boxed{3b^2 + 18b - 48} \]

What is the product of the polynomials (x2y+2)(x2−y) ?(1 point)
x^3y−x^2y^2+2x^2−2y
x^4y−xy+x^2y
x^4y+x^2−2y
x^4y−x^2y^2+2x^2−2y

To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we will use the distributive property (also known as the FOIL method for binomials).

We distribute each term in the first polynomial by each term in the second polynomial:

\[ (x^2y + 2)(x^2 - y) = x^2y(x^2) + x^2y(-y) + 2(x^2) + 2(-y) \]

Calculating each term gives us:

  1. \(x^2y \cdot x^2 = x^4y\)
  2. \(x^2y \cdot (-y) = -x^2y^2\)
  3. \(2 \cdot x^2 = 2x^2\)
  4. \(2 \cdot (-y) = -2y\)

Now, combine all these terms:

\[ x^4y - x^2y^2 + 2x^2 - 2y \]

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

\[ \boxed{x^4y - x^2y^2 + 2x^2 - 2y} \]

Which expression is equivalent to x3(2+y5) ?(1 point)

x^3+2+y^5
2x^3+x^3y^5
2x^3+y^5
x^3+x^3+xy+xy+xy

To determine which expression is equivalent to \( x^3(2 + y^5) \), we can distribute \( x^3 \) into the parentheses:

\[ x^3(2 + y^5) = x^3 \cdot 2 + x^3 \cdot y^5 = 2x^3 + x^3y^5 \]

Thus, the expression simplifies to:

\[ 2x^3 + x^3y^5 \]

Now, among the options provided, the equivalent expression is:

\[ \boxed{2x^3 + x^3y^5} \]

Which of the following responses demonstrates that polynomials form a closed system under multiplication?(1 point)

x^2+2x+1 x
(x2/2)(1/x)
(x^2+1)(x−1/2)
(x−−√)(x+1)

To determine which response demonstrates that polynomials form a closed system under multiplication, we need to remember that a set is closed under an operation if performing that operation on members of the set always produces a member of the same set.

Let's analyze each option:

  1. \(x^2 + 2x + 1 \cdot x\):

    • This simplifies to \(x^3 + 2x^2 + x\), which is a polynomial. Thus, it shows closure under multiplication.

  2. \((x^2/2)(1/x)\):

    • This simplifies to \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\), which is not a polynomial since it has a rational coefficient and can be expressed as a non-integral power of \(x\) (specifically, \(x^{-1}\)). This does not demonstrate closure under multiplication.

  3. \((x^2 + 1)(x - 1/2)\):

    • This is a product of two polynomials and will result in another polynomial. Thus, it shows closure under multiplication.

  4. \((x - \sqrt{x})(x + 1)\):

    • While both \(x - \sqrt{x}\) and \(x + 1\) look like polynomials, the term \(\sqrt{x}\) is not a polynomial, as it can be expressed as \(x^{1/2}\), which is a non-integer exponent. Therefore, this does not represent a polynomial.

Conclusion: The options that demonstrate polynomials form a closed system under multiplication are:

  1. \(x^2 + 2x + 1 \cdot x\)
  2. \((x^2 + 1)(x - 1/2)\)

If we need to choose one, the first option is a straightforward case of polynomial multiplication:

\[ \boxed{x^2 + 2x + 1 \cdot x} \]

This confirms polynomials' closure under multiplication effectively.

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (12x2−3)(4y3+5x2) (1 point)

x^2y^3+5/x^4−8y^3−15x^2
√x^2y^3+4x^4−15x^2
5/2x^2y^3+2x^4−8y^3+15x^2
2x^2y^3+5/2x^4−12y^3−15x^2

To find the product of the polynomials \((12x^2 - 3)(4y^3 + 5x^2)\), we will use the distributive property (also known as the FOIL method for binomials).

Let's distribute each term in the first polynomial by each term in the second polynomial:

\[ (12x^2 - 3)(4y^3 + 5x^2) = 12x^2(4y^3) + 12x^2(5x^2) - 3(4y^3) - 3(5x^2) \]

Calculating each term gives us:

  1. \(12x^2 \cdot 4y^3 = 48x^2y^3\)
  2. \(12x^2 \cdot 5x^2 = 60x^4\)
  3. \(-3 \cdot 4y^3 = -12y^3\)
  4. \(-3 \cdot 5x^2 = -15x^2\)

Now, combine all these terms:

\[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]

Thus, the product of \((12x^2 - 3)(4y^3 + 5x^2)\) is:

\[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

Now, let's arrange it in a standard polynomial form:

\[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]

Now we can compare this result against the provided options:

  1. \(x^2y^3+5/x^4−8y^3−15x^2\)
  2. \(\sqrt{x^2y^3}+4x^4−15x^2\)
  3. \(5/2x^2y^3+2x^4−8y^3+15x^2\)
  4. \(2x^2y^3+5/2x^4−12y^3−15x^2\)

Since none of these match directly with the calculated product, we can conclude that none of the provided options are correct based on our multiplication process.

However, if we had to choose the option that demonstrates a polynomial or its derived properties, you should ensure it maintains integer coefficients and polynomial format, while also confirming the closure property.

As such, if you ever face similar options, check if polynomials yield consistent polynomial forms as we've computed.

Question #4 is (x^2+1)(x-1/2)
Question #5 is 2x^2y^3+5/4x^-12y^3-15x^2

Thank you for providing the specific products for Questions #4 and #5. Let's clarify their relevance concerning the closure property of polynomials and determine their validity.

Question #4: \((x^2 + 1)(x - \frac{1}{2})\)

We will multiply these polynomials:

\[ (x^2 + 1)(x - \frac{1}{2}) = x^2 \cdot x + x^2 \cdot \left(-\frac{1}{2}\right) + 1 \cdot x + 1 \cdot \left(-\frac{1}{2}\right) \]

Calculating each term yields:

  1. \(x^2 \cdot x = x^3\)
  2. \(x^2 \cdot \left(-\frac{1}{2}\right) = -\frac{1}{2}x^2\)
  3. \(1 \cdot x = x\)
  4. \(1 \cdot \left(-\frac{1}{2}\right) = -\frac{1}{2}\)

Combining these results, we have:

\[ x^3 - \frac{1}{2}x^2 + x - \frac{1}{2} \]

This expression is indeed a polynomial, thus demonstrating that the product of two polynomials yields another polynomial. Therefore, this demonstrates closure under multiplication.

Question #5: \(2x^2y^3 + \frac{5}{4}x^{-12}y^3 - 15x^2\)

This expression contains the term \(\frac{5}{4}x^{-12}y^3\). The presence of \(x^{-12}\) indicates that this is not a polynomial as polynomials can only have non-negative integral exponents. Hence, this expression does not satisfy the closure property since not every term is a polynomial.

Conclusion:

  • For Question #4 \((x^2 + 1)(x - \frac{1}{2})\), the product demonstrates that the set of polynomials is closed under multiplication.
  • For Question #5, due to the \(x^{-12}\) term, this does not remain a polynomial, thus it does not confirm the closure property.

If further clarification or exploration is needed, feel free to ask!