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According to the Distributive Property, which expression is equivalent to 4x3+8x2+12x

?(1 point)
Responses

(2x2+4)(2x−3)
left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x minus 3 right parenthesis

4x(4x+5x)
4 x left parenthesis 4 x plus 5 x right parenthesis

4x(x2+2x+3)
4 x left parenthesis x squared plus 2 x plus 3 right parenthesis

(2x2+4)(2x2+3)

11 answers

To find an expression equivalent to \(4x^3 + 8x^2 + 12x\) using the Distributive Property, we can factor out the greatest common factor from all the terms in the expression.

The terms in the expression are \(4x^3\), \(8x^2\), and \(12x\). The greatest common factor is \(4x\).

Now, we can factor out \(4x\):

\[ 4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3) \]

Now that we have factored the original expression, we can check which of the given responses matches this.

The correct response from the options provided is:

4x(x² + 2x + 3).

So, the answer is:

4x(x² + 2x + 3).

Which of the following simplifies to the same expression as 4x2−3x2+5x−24
?(1 point)
Responses

(x+8)(x−3)
left parenthesis x plus 8 right parenthesis left parenthesis x minus 3 right parenthesis

4x(x−6)
4 x left parenthesis x minus 6 right parenthesis

2(x−12)
2 left parenthesis x minus 12 right parenthesis

(x+6)(x−4)

Let's first simplify the expression \(4x^2 - 3x^2 + 5x - 24\).

  1. Combine like terms for the \(x^2\) terms: \[ 4x^2 - 3x^2 = 1x^2 \quad \text{or} \quad x^2 \]

  2. The expression now looks like: \[ x^2 + 5x - 24 \]

Next, we can factor \(x^2 + 5x - 24\). We need to find two numbers that multiply to \(-24\) (the constant term) and add to \(5\) (the coefficient of \(x\)).

The numbers \(8\) and \(-3\) satisfy these conditions:

  • \(8 \times -3 = -24\)
  • \(8 + (-3) = 5\)

Therefore, we can factor the expression as: \[ (x + 8)(x - 3) \]

Now, let’s check which of the provided responses matches this factorization:

  1. (x + 8)(x - 3) — This matches our factorization.
  2. 4x(x - 6) — This does not match.
  3. 2(x - 12) — This does not match.
  4. (x + 6)(x - 4) — This does not match.

The expression \(4x^2 - 3x^2 + 5x - 24\) simplifies to \( (x + 8)(x - 3) \).

So, the answer is: (x + 8)(x - 3).

9 of 179 of 17 Items

Question
Add the polynomial expressions 5z−2z2−4z+5
and 6z2+3−7z
.(1 point)
Responses

4z2−6z+8
4 z squared minus 6 z plus 8

9z2−11z+8
9 z squared minus 11 z plus 8

4z2−z+8
4 z squared minus z plus 8

11z2+z−2

To add the polynomial expressions \(5z - 2z^2 - 4z + 5\) and \(6z^2 + 3 - 7z\), we will first combine the terms from both expressions.

  1. Start by rewriting both expressions:

    • The first expression: \(5z - 2z^2 - 4z + 5\)
    • The second expression: \(6z^2 + 3 - 7z\)

  2. Combine like terms from the first expression: \[ 5z - 4z = 1z \quad \text{so the first expression simplifies to} \quad -2z^2 + z + 5 \]

  3. Now, rewrite the first expression with combined terms: \[ -2z^2 + z + 5 \]

  4. Now add the second expression: \[ -2z^2 + z + 5 + 6z^2 + 3 - 7z \]

  5. Combine like terms:

    • For \(z^2\) terms: \(-2z^2 + 6z^2 = 4z^2\)
    • For \(z\) terms: \(z - 7z = -6z\)
    • For the constant terms: \(5 + 3 = 8\)

  6. Putting it all together: \[ 4z^2 - 6z + 8 \]

Now, we can look at your response choices:

  • 4z² - 6z + 8 — This matches our calculation.
  • 9z² - 11z + 8 — This does not match.
  • 4z² - z + 8 — This does not match.
  • 11z² + z - 2 — This does not match.

Thus, the answer is: 4z² - 6z + 8.

Which of the following correctly simplifies (9x2+17x+13)−(2x2+12x+11)
?(1 point)
Responses

7x2+29x+24
7 x squared plus 29 x plus 24

7x2+5x+2
7 x squared plus 5 x plus 2

9x2+17x+13−2x2−12x−11
9 x squared plus 17 x plus 13 minus 2 x squared minus 12 x minus 11

12x2+2

To simplify the expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), we need to distribute the negative sign across the second polynomial and then combine like terms.

  1. Start with the given expression: \[ (9x^2 + 17x + 13) - (2x^2 + 12x + 11) \]

  2. Distributing the negative sign to the second polynomial: \[ 9x^2 + 17x + 13 - 2x^2 - 12x - 11 \]

  3. Now, combine like terms:

    • For the \(x^2\) terms: \(9x^2 - 2x^2 = 7x^2\)
    • For the \(x\) terms: \(17x - 12x = 5x\)
    • For the constant terms: \(13 - 11 = 2\)

  4. Putting it all together, we have: \[ 7x^2 + 5x + 2 \]

Now, let's check the responses:

  • 7x² + 29x + 24 — This does not match.
  • 7x² + 5x + 2 — This matches our calculation.
  • 9x² + 17x + 13 - 2x² - 12x - 11 — This represents the same expression but is not simplified.
  • 12x² + 2 — This does not match.

Therefore, the correct answer is: 7x² + 5x + 2.

Multiply the polynomials (3x−1)(2x+4)
.(2 points)
x2+
x+

To multiply the polynomials \( (3x - 1)(2x + 4) \), we can use the distributive property (also known as the FOIL method for binomials).

  1. First: Multiply the first terms in each binomial: \[ 3x \cdot 2x = 6x^2 \]

  2. Outer: Multiply the outer terms: \[ 3x \cdot 4 = 12x \]

  3. Inner: Multiply the inner terms: \[ -1 \cdot 2x = -2x \]

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

Now, combine all the results: \[ 6x^2 + 12x - 2x - 4 \]

Combine like terms: \[ 6x^2 + (12x - 2x) - 4 = 6x^2 + 10x - 4 \]

Now, we can write the final result as: \[ 6x^2 + 10x - 4 \]

So to fill in the blanks:

  • \(x^2 + 10x - 4\)

Final answer: \[ 6x^2 + 10x - 4 \]

(Note: Since the formatting of the answer requested specifically "x^2 +", it seems you might want components separated, so it would fit as \(x^2 + 10x - 4\) accordingly).

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

(x)(12x)
left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis

(x−−√)(6x ⋅−3 )
open paren square root of x close paren times open paren 6 x times negative 3 close paren

3x24−1
Start Fraction 3 x squared over 4 minus 1 End Fraction

14x3⋅5

To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to find an expression where the product of two polynomial expressions is also a polynomial.

Analyzing each option:

  1. \((x)(12x)\):

    • The multiplication of \(x\) (a polynomial of degree 1) and \(12x\) (a polynomial of degree 1) results in \(12x^2\), which is also a polynomial.
    • This shows closure under multiplication.

  2. \((\sqrt{x})(6x \cdot -3)\):

    • \(\sqrt{x}\) is not a polynomial (it is a root) so this expression does not qualify as a polynomial multiplication and doesn't demonstrate closure under multiplication of polynomials.

  3. \(\frac{3x^2}{4} - 1\):

    • This is a polynomial expression itself, but it is not a multiplication of two polynomials, so it does not demonstrate the closure property.

  4. \(14x^3 \cdot 5\):

    • This multiplication results in \(70x^3\), which is a polynomial. However, because one of the factors is a constant (which is technically a polynomial of degree 0), this also shows closure.

Conclusion: Both options 1 and 4 demonstrate closure under multiplication of polynomials, but option 1 (\((x)(12x)\)) is more straightforward as it consists purely of polynomials.

Thus, the response that clearly shows that polynomials form a closed system under multiplication is: \((x)(12x)\).