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Question

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

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

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

b2+18b−36
b squared plus 18 b minus 36

3b2+18b−48

9 answers

To multiply the polynomial \((b + 8)(3b - 6)\), we can use the distributive property (also known as the FOIL method for binomials) to simplify the expression.

  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 we add all these results together:

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

Combine the like terms \(-6b\) and \(24b\):

\[ 3b^2 + 18b - 48 \]

Thus, the simplified polynomial is:

\[ 3b^2 + 18b - 48 \]

So the correct response is:

3b² + 18b - 48.

Multiplying Polynomials Quick Check
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Question
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+x2−2y
x superscript 4 baseline y plus x squared minus 2 y

x4y−x2y2+2x2−2y
x superscript 4 baseline 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
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To find the product of the polynomials \((x^2y + 2)(x^2 - y)\), we'll use the distributive property (FOIL method) for binomials.

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

  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 of these results together:

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

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

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

The correct response is:

x⁴y - x²y² + 2x² - 2y.

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

2x3+x3y5
2 x cubed plus x cubed y superscript 5 baseline

2x3+y5
2 x cubed plus y superscript 5 baseline

x3+2+y5
x cubed plus 2 plus y superscript 5 baseline

x3+x3+xy+xy+xy

To simplify the expression \( x^3(2 + y^5) \), you can distribute \( x^3 \) to both terms inside the parentheses:

  1. Multiply \( x^3 \) by \( 2 \): \[ x^3 \cdot 2 = 2x^3 \]

  2. Multiply \( x^3 \) by \( y^5 \): \[ x^3 \cdot y^5 = x^3y^5 \]

Now, combine the results together: \[ 2x^3 + x^3y^5 \]

Thus, the expression \( x^3(2 + y^5) \) is equivalent to:

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

The correct response is:

2x³ + x³y⁵.

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

(x22)(1x)
left parenthesis Start Fraction x squared over 2 End Fraction right parenthesis left parenthesis Start Fraction 1 over x End Fraction right parenthesis

x2+2x+1
x squared plus 2 x plus 1

(x2+1)(x−12)
left parenthesis x squared plus 1 right parenthesis left parenthesis x minus Start Fraction 1 over 2 End Fraction right parenthesis

(x−−√)(x+1)

To demonstrate that polynomials form a closed system under multiplication, we need to show that when two polynomials are multiplied together, the result is also a polynomial.

Let's analyze each of the given responses:

  1. \((x^2)(\frac{1}{x})\):

    • This expression results in \(x^{2-1} = x^1\), which is a polynomial. However, it includes a rational function \( \frac{1}{x} \), which is not a polynomial. Thus, this does not demonstrate closure.

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

    • This is already a polynomial. However, it doesn't show multiplication or closure.

  3. \((x^2 + 1)(x - \frac{1}{2})\):

    • This is a multiplication of two polynomials. The result of this multiplication will still be a polynomial since multiplying polynomials yields a polynomial. This response shows closure.

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

    • The term \(\sqrt{x}\) is not a polynomial, as polynomials cannot have fractional or negative exponents. Therefore, this does not demonstrate closure.

The correct response that demonstrates that polynomials form a closed system under multiplication is:

\((x^2 + 1)(x - \frac{1}{2})\).

Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.

(12x2−3)(4y3+5x2)

(1 point)
Responses

52x2y3+2x4−8y3+15x2
Start Fraction 5 over 2 End Fraction x squared y cubed plus 2 x superscript 4 baseline minus 8 y cubed plus 15 x squared

2x2y3+52x4−12y3−15x2
2 x squared y cubed plus Start Fraction 5 over 2 End Fraction x superscript 4 baseline minus 12 y cubed minus 15 x squared

x2y3+5x4−8y3−15x2
x squared y cubed plus Start Fraction 5 over x superscript 4 baseline End Fraction minus 8 y cubed minus 15 x squared

x2−−√y3+4x4−15x2

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

  1. Distribute \(12x^2\) to both terms in the second polynomial:

    • \(12x^2 \cdot 4y^3 = 48x^2y^3\)
    • \(12x^2 \cdot 5x^2 = 60x^4\)

  2. Distribute \(-3\) to both terms in the second polynomial:

    • \(-3 \cdot 4y^3 = -12y^3\)
    • \(-3 \cdot 5x^2 = -15x^2\)

Now add all these results together: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]

Rearranging the terms, we get: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

Thus, the product that demonstrates the closure property of multiplication of polynomials is: \[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]

None of the options provided exactly match this final expression, so if there’s a way to correct or adjust that in your context or if I misunderstood the responses offered, please let me know!