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

Question

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

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

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

4x(x2+2x+3)

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11 answers
(click or scroll down)

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Answer 1

Bot GPT 4-o mini answered
1 month ago

To find the expression that is equivalent to \(4x^3 + 8x^2 + 12x\) using the Distributive Property, let's factor out the common factor.

1. Identify the common factor in the expression:
\[
4x^3 + 8x^2 + 12x
\]
The common factor here is \(4x\).

2. Factor out \(4x\):
\[
4x(x^2 + 2x + 3)
\]

3. Now we look for an answer choice that matches \(4x(x^2 + 2x + 3)\).

The expression \(4x(x^2 + 2x + 3)\) corresponds directly with the option:
- \(4x(x^2 + 2x + 3)\)

Thus, the correct answer is:
**4x(x^2 + 2x + 3)**.

Answer 2

anonymous answered
1 month ago

Which of the following simplifies to the same expression as 4x2−3x2+5x−24

?(1 point)
Responses

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

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

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

(x+6)(x−4)

Answer 3

Bot GPT 4-o mini answered
1 month ago

To determine which expression simplifies to \(4x^2 - 3x^2 + 5x - 24\), we first simplify the provided expression:

1. Combine like terms in \(4x^2 - 3x^2 + 5x - 24\):
\[
(4x^2 - 3x^2) + 5x - 24 = 1x^2 + 5x - 24 = x^2 + 5x - 24.
\]

Next, we need to factor \(x^2 + 5x - 24\) and check if any of the provided options are equivalent to this expression.

To factor \(x^2 + 5x - 24\), we look for two numbers that multiply to \(-24\) and add to \(5\). The numbers \(8\) and \(-3\) work because:
\[
8 \times (-3) = -24 \quad \text{and} \quad 8 + (-3) = 5.
\]

Thus, we can factor the expression:
\[
x^2 + 5x - 24 = (x + 8)(x - 3).
\]

Now, let’s compare our factorization to the provided options. The correct matching expression is:
- \((x + 8)(x - 3)\).

Therefore, the expression that simplifies to the same expression as \(4x^2 - 3x^2 + 5x - 24\) is:
**(x + 8)(x - 3)**.

Answer 4

anonymous answered
1 month ago

Add the polynomial expressions 5z−2z2−4z+5

and 6z2+3−7z
.(1 point)
Responses

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

4z2−z+8
4 z squared minus z plus 8

4z2−6z+8
4 z squared minus 6 z plus 8

11z2+z−2

Answer 5

Bot GPT 4-o mini answered
1 month ago

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

1. First, let's rewrite each expression and then add them together:

**First expression:**
\[
5z - 2z^2 - 4z + 5
\]

Combine the like terms \(5z\) and \(-4z\):
\[
= -2z^2 + (5z - 4z) + 5 = -2z^2 + 1z + 5
\]

So the first expression simplifies to:
\[
-2z^2 + z + 5
\]

**Second expression:**
\[
6z^2 + 3 - 7z
\]

Rearranging gives us:
\[
6z^2 - 7z + 3
\]

2. Now, we can add both simplified expressions together:
\[
(-2z^2 + z + 5) + (6z^2 - 7z + 3)
\]

Combine like terms:
- For \(z^2\) terms: \(-2z^2 + 6z^2 = 4z^2\)
- For \(z\) terms: \(1z - 7z = -6z\)
- For constant terms: \(5 + 3 = 8\)

So the combined expression becomes:
\[
4z^2 - 6z + 8
\]

Now let's identify the correct option:
**4z² - 6z + 8**
This is the answer!

Therefore, the answer is:
**4z² - 6z + 8**.

Answer 6

anonymous answered
1 month ago

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

12x2+2
12 x squared plus 2

9x2+17x+13−2x2−12x−11

Answer 7

Bot GPT 4-o mini answered
1 month ago

To simplify the expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), we will first distribute the negative sign in front of the second polynomial, and then combine like terms.

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

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

3. Now, combine the like terms:
- For \(x^2\) terms:
\[
9x^2 - 2x^2 = 7x^2
\]
- For \(x\) terms:
\[
17x - 12x = 5x
\]
- For constant terms:
\[
13 - 11 = 2
\]

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

Now, let's check the response options:
- **7x² + 29x + 24**
- **7x² + 5x + 2** (This is correct!)
- **12x² + 2**
- **9x² + 17x + 13 − 2x² − 12x − 11** (This is the expression written out again but not simplified.)

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

Answer 8

anonymous answered
1 month ago

Multiply the polynomials (3x−1)(2x+4)

.(2 points)
x2+
x+

Answer 9

Bot GPT 4-o mini answered
1 month ago

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

1. **First, multiply the first terms:**
\[
3x \cdot 2x = 6x^2
\]

2. **Next, multiply the outer terms:**
\[
3x \cdot 4 = 12x
\]

3. **Then, multiply the inner terms:**
\[
-1 \cdot 2x = -2x
\]

4. **Finally, multiply the last terms:**
\[
-1 \cdot 4 = -4
\]

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

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

So the final result is:
\[
6x^2 + 10x - 4
\]

Thus, you can fill in the blanks as follows:
\[
x^2 + 10x - 4
\]

So:
- \( x^2 = 6x^2 \) (which you will note as "6").
- \( x = 10 \).
- The constant is \(-4\).

Final answer:
- \(6x^2 + 10x - 4\) (with individual values being filled as described above).

Answer 10

anonymous answered
1 month ago

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

Responses

14x3⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5

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

(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

Answer 11

Bot GPT 4-o mini answered
1 month ago

To show that polynomials form a closed system under multiplication, we need to demonstrate that the product of any two polynomial expressions is also a polynomial.

Let’s evaluate each of the listed responses:

1. **\(14x^3 \cdot 5\)**
- This is the multiplication of a polynomial \(14x^3\) (with degree 3) and a constant \(5\). The result will be \(70x^3\), which is a polynomial.

2. **\(\frac{3x^2}{4} - 1\)**
- This expression is a polynomial (specifically, a rational polynomial). However, it does not demonstrate multiplication.

3. **\((x)(\frac{1}{2} x)\)**
- This is the multiplication of two polynomials: \(x\) and \(\frac{1}{2} x\), which yields \(\frac{1}{2} x^2\), a polynomial.

4. **\((\sqrt{x})(6x \cdot -3)\)**
- The term \(\sqrt{x}\) is not a polynomial, since polynomials only include non-negative integer exponents. Thus, this expression does not demonstrate closure.

The first three responses contain polynomial expressions, and their products result in a polynomial. However, the best response demonstrating that polynomials form a closed system under multiplication is:

**\(14x^3 \cdot 5\)**, because it clearly shows two polynomial terms being multiplied and producing another polynomial result.

Answer:
**14x³ ⋅ 5** (which yields \(70x^3\), a polynomial).