Ask a New Question
Menu

Add the polynomial expressions 5z−2z^2−4z+5 and 6z^2+3−7z

Responses:
4z2−6z+8
11z2+z−2
9z2−11z+8
4z2−z+8

11 answers

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

  1. Combine the first polynomial: \[ 5z - 4z - 2z^2 + 5 = (5z - 4z) + (-2z^2) + 5 = 1z - 2z^2 + 5 = -2z^2 + z + 5 \]

  2. Write the second polynomial as is: \[ 6z^2 - 7z + 3 \]

  3. Now, we add the two results: \[ (-2z^2 + z + 5) + (6z^2 - 7z + 3) \] 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 get: \[ 4z^2 - 6z + 8 \]

Thus, the correct response is: 4z² - 6z + 8.

Which of the following correctly simplifies (9x^2+17x+13)−(2x^2+12x+11)
Responses:
12x2+2
7x2+29x+24
9x2+17x+13−2x2−12x−11
7x2+5x+2

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

  1. Distribute the negative sign: \[ (9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 9x^2 + 17x + 13 - 2x^2 - 12x - 11 \]

  2. Combine 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 get: \[ 7x^2 + 5x + 2 \]

Thus, the correct response is: 7x² + 5x + 2.

Multiply the polynomials (3x−1)(2x+4)

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

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

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

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

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

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

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

Thus, the product of the polynomials \((3x - 1)(2x + 4)\) is: \[ \boxed{6x^2 + 10x - 4} \]

Which of the following responses shows that polynomials form a closed system under multiplication?(1 point)
Responses:
1/4x^3⋅5
3x^2/4−1
(x)(1/2x)
(x−−√)(6x ⋅−3 )

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

Let's analyze each of the response options:

  1. \( \frac{1}{4}x^3 \cdot 5 \)
    This results in \( \frac{5}{4}x^3 \), which is a polynomial.

  2. \( \frac{3x^2}{4} - 1 \)
    This expression is a polynomial expression, but it does not perform a multiplication between two polynomials.

  3. \( (x)(\frac{1}{2}x) \)
    This results in \( \frac{1}{2}x^2 \), which is a polynomial.

  4. \( (\sqrt{x})(6x \cdot -3) \)
    The term \(\sqrt{x}\) is not a polynomial because polynomials cannot have fractional or irrational exponents. Thus, this expression does not show closure under multiplication.

The correct responses that demonstrate polynomials forming a closed system under multiplication are:

  • \( \frac{1}{4}x^3 \cdot 5 \)
  • \( (x)(\frac{1}{2}x) \)

However, if you are looking for a single response, then \( \frac{1}{4}x^3 \cdot 5 \) and \( (x)(\frac{1}{2}x) \) both are valid examples showing closure under multiplication. If only one option is to be chosen, either would suffice, but typically, \( \frac{1}{4}x^3 \cdot 5 \) is more straightforward.

Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences, explain your result.
Expression 1: y⋅3x
Expression 2: 3xy

To prove that the expressions \( y \cdot 3x \) and \( 3xy \) are equivalent using the Commutative Property, we can rearrange the factors in \( y \cdot 3x \):

\[ y \cdot 3x = 3 \cdot y \cdot x = 3xy \]

By the Commutative Property of multiplication, the order of the factors in a product does not change the result. Thus, the expressions \( y \cdot 3x \) and \( 3xy \) represent the same quantity, proving their equivalence.

Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2.
Expression 1: 22r−13
Expression 2: 19r+(3r−2)−11

To demonstrate that the expressions \(22r - 13\) and \(19r + (3r - 2) - 11\) are equivalent using the Associative Property, we start by simplifying Expression 2.

Expression 2:

\[ 19r + (3r - 2) - 11 \]

First, we can simplify the expression inside the parentheses: \[ = 19r + 3r - 2 - 11 \]

Next, combine the like terms: \[ 19r + 3r - 2 - 11 = (19r + 3r) + (-2 - 11) = 22r - 13 \]

Now we have: \[ 19r + (3r - 2) - 11 = 22r - 13 \]

Conclusion:

Since we have shown that \(19r + (3r - 2) - 11\) simplifies to \(22r - 13\), we conclude that both expressions are equivalent. The Associative Property allows us to regroup terms without changing the result, which we used to combine \(19r\) and \(3r\) effectively.