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

Question

?(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)

Answer 1

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).

Answer 2

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)

Answer 3

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

1. Combine like terms for the $x^2$ terms: $$4x^2-3x^2=1x^2\text{or}{x}^2$$

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).

Answer 4

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

Answer 5

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: $$5z-4z=1z\text{so the first expression simplifies to}-2z^2+z+5$$

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.

Answer 6

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

Answer 7

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.

Answer 8

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

Answer 9

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).

Answer 10

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

Answer 11

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: $(x)(12x)$.