?(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)
11 answers
To find the expression that is equivalent to \(4x^3 + 8x^2 + 12x\) using the Distributive Property, let's factor out the common factor.
Identify the common factor in the expression: \[ 4x^3 + 8x^2 + 12x \] The common factor here is \(4x\).
Factor out \(4x\): \[ 4x(x^2 + 2x + 3) \]
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:
Thus, the correct answer is: 4x(x^2 + 2x + 3).
To determine which expression simplifies to \(4x^2 - 3x^2 + 5x - 24\), we first simplify the provided expression:
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:
Therefore, the expression that simplifies to the same expression as \(4x^2 - 3x^2 + 5x - 24\) is: (x + 8)(x - 3).
To add the polynomial expressions \(5z - 2z^2 - 4z + 5\) and \(6z^2 + 3 - 7z\), we will combine like terms step by step.
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: \[ 6z^2 - 7z + 3 \]
Now, we can add both simplified expressions together: \[ (-2z^2 + z + 5) + (6z^2 - 7z + 3) \]
Combine like terms:
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.
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.
Start with the expression: \[ (9x^2 + 17x + 13) - (2x^2 + 12x + 11) \]
Distributing the negative sign: \[ 9x^2 + 17x + 13 - 2x^2 - 12x - 11 \]
Now, combine the like terms:
Putting it all together, we have: \[ 7x^2 + 5x + 2 \]
Now, let's check the response options:
Therefore, the correct choice is: 7x² + 5x + 2.
To multiply the polynomials \((3x - 1)(2x + 4)\), we can use the distributive property (often referred to as the FOIL method for binomials).
First, multiply the first terms:
\[
3x \cdot 2x = 6x^2
\]
Next, multiply the outer terms:
\[
3x \cdot 4 = 12x
\]
Then, multiply the inner terms:
\[
-1 \cdot 2x = -2x
\]
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:
Final answer:
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:
\(14x^3 \cdot 5\)
\(\frac{3x^2}{4} - 1\)
\((x)(\frac{1}{2} x)\)
\((\sqrt{x})(6x \cdot -3)\)
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).
