Question

Question

Question

According to the Distributive Property, which expression is equivalent to 4x3+8x2+12x
?(1 point)
Responses

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)
left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x squared plus 3 right parenthesis

(2x2+4)(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 an equivalent expression for \( 4x^3 + 8x^2 + 12x \) using the Distributive Property, we first look for a common factor in each term of the expression.

The given expression is:
\[ 4x^3 + 8x^2 + 12x \]

We can factor out the greatest common factor, which is \( 4x \):
\[ 4x(x^2 + 2x + 3) \]

Now we need to compare this expression to the options provided:

1. \( 4x(4x + 5x) \) — This simplifies to \( 4x(9x) \), which does not match.
2. \( 4x(x^2 + 2x + 3) \) — This is the expression we arrived at, and it is equivalent.
3. \( (2x^2 + 4)(2x^2 + 3) \) — This does not match our expression when expanded.
4. \( (2x^2 + 4)(2x - 3) \) — This also does not match when expanded.

Thus, the correct answer is:
\[ \boxed{4x(x^2 + 2x + 3)} \]

Answer 2

x answered
1 month ago

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

?(1 point)
Responses

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

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

(x+6)(x−4)

Answer 3

Bot GPT 4-o mini answered
1 month ago

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

1. Combine like terms for the \( x^2 \) terms:
\[
4x^2 - 3x^2 = x^2
\]
So, the expression simplifies to:
\[
x^2 + 5x - 24
\]

Now we need to 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 the linear term). The numbers are \(8\) and \(-3\).

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

Now we will compare this factored expression with the options provided:

1. \( 2(x - 12) \) — This does not match our factored expression.
2. \( 4x(x - 6) \) — This also does not match.
3. \( (x + 8)(x - 3) \) — This matches the expression we obtained.
4. \( (x + 6)(x - 4) \) — This does not match when expanded.

The expression that simplifies to \( 4x^2 - 3x^2 + 5x - 24 \) is:
\[ \boxed{(x + 8)(x - 3)} \]

Answer 4

x answered
1 month ago

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

and 6z^2+3−7z
.(1 point)
Responses

11z^2+z−2
11 z squared plus z minus 2

4z^2−z+8
4 z squared minus z plus 8

4z^2−6z+8
4 z squared minus 6 z plus 8

9z^2−11z+8
9 z squared minus 11 z plus 8

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 first simplify each expression.

### Step 1: Simplify the first polynomial
\[
5z - 2z^2 - 4z + 5
\]
We can combine like terms (the \( z \) terms):
\[
(5z - 4z) - 2z^2 + 5 = z - 2z^2 + 5
\]
Thus, the first expression simplifies to:
\[
-2z^2 + z + 5
\]

### Step 2: Simplify the second polynomial
\[
6z^2 + 3 - 7z
\]
We can rearrange this as:
\[
6z^2 - 7z + 3
\]

### Step 3: Add the simplified expressions
Now we add the two simplified polynomials together:
\[
(-2z^2 + z + 5) + (6z^2 - 7z + 3)
\]

Now, let’s combine like terms:

- **For \( z^2 \) terms:**
\(-2z^2 + 6z^2 = 4z^2\)

- **For \( z \) terms:**
\(z - 7z = -6z\)

- **For constant terms:**
\(5 + 3 = 8\)

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

Thus, the result is:
\[
\boxed{4z^2 - 6z + 8}
\]

Answer 6

x answered
1 month ago

Which of the following correctly simplifies (9x^2+17x+13)−(2x^2+12x+11)

?(1 point)
Responses

7x^2+29x+24
7 x squared plus 29 x plus 24

12x^2+2
12 x squared plus 2

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

7x^2+5x+2

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 across the second polynomial:

\[
= 9x^2 + 17x + 13 - 2x^2 - 12x - 11
\]

Now we can rearrange and combine like terms. Grouping the \(x^2\), \(x\), and constant terms together:

1. **For \(x^2\) terms:**
\[
9x^2 - 2x^2 = 7x^2
\]

2. **For \(x\) terms:**
\[
17x - 12x = 5x
\]

3. **For constant terms:**
\[
13 - 11 = 2
\]

Now putting it all together, we have:

\[
7x^2 + 5x + 2
\]

Thus, the simplified expression is:
\[
\boxed{7x^2 + 5x + 2}
\]

Answer 8

x 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 will use the distributive property (also known as the FOIL method for binomials).

### Steps:

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

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:
\[
-1 \cdot 4 = -4
\]

### Combine the results:
Now combine all the terms together:
\[
6x^2 + 12x - 2x - 4
\]

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

### Final expression:
Thus, the final result of the multiplication is:
\[
6x^2 + 10x - 4
\]

### Format:
In the requested format:
- Coefficient of \(x^2\): **6**
- Coefficient of \(x\): **10**
- Constant term: **-4**

So, the answer is:
\[
(6)x^2 + (10)x + (-4)
\]

Complete answer:
\[
\boxed{(6)x^2 + (10)x + (-4)}
\]

Answer 10

x answered
1 month ago

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

Responses

1/4x^3 ⋅5
Start Fraction 1 over 4 x cubed End Fraction times 5

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

3x^2/4−1
Start Fraction 3 x squared over 4 minus 1 End Fraction

(x)(1/2 x)

Answer 11

Bot GPT 4-o mini answered
1 month ago

To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial. Let's evaluate the responses given:

1. **\( \frac{1}{4}x^3 \cdot 5 \)**
- This is a multiplication of a polynomial \( \frac{1}{4}x^3 \) (which is a polynomial) and a constant \( 5 \). The result is \( \frac{5}{4}x^3 \), which is still a polynomial.

2. **\( (\sqrt{x})(6x \cdot -3) \)**
- Here, \( \sqrt{x} \) is not a polynomial (it is a radical expression). The product of a non-polynomial and a polynomial does not guarantee that the result is a polynomial.

3. **\( \frac{3x^2}{4} - 1 \)**
- This expression represents a polynomial (even though it includes a fraction), but it does not illustrate the multiplication of two polynomials.

4. **\( (x)(\frac{1}{2}x) \)**
- This represents the multiplication of the polynomial \( x \) and \( \frac{1}{2}x \). The result is \( \frac{1}{2}x^2 \), which is also a polynomial.

### Conclusion
The responses that show that polynomials form a closed system under multiplication are:

- \( \frac{1}{4}x^3 \cdot 5 \) (produces a polynomial)
- \( (x)(\frac{1}{2}x) \) (produces a polynomial)

Since we need to choose one response that clearly shows closure under multiplication, we can choose either one, but the clearest example is:

\[
\boxed{(x)(\frac{1}{2} x)}
\]

This example captures the essence of closure under polynomial multiplication succinctly.