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What are the different types of terms that appear in the polynomial expression?

2(x2−x)+6x−3−7x
Responses:
Constant.
Linear.
Quadratic.
Cubic.

11 answers

In the polynomial expression \( 2(x^2 - x) + 6x - 3 - 7x \), we can simplify it first to identify the types of terms that appear.

  1. Distributing the 2: \[ 2(x^2 - x) = 2x^2 - 2x \]

  2. Now, substituting this back into the expression: \[ 2x^2 - 2x + 6x - 3 - 7x \]

  3. Combining like terms:

    • The \(x\) terms: \(-2x + 6x - 7x = -3x\)
    • The expression simplifies to: \[ 2x^2 - 3x - 3 \]

Now, let's analyze the terms in the simplified polynomial \( 2x^2 - 3x - 3 \):

  • Quadratic term: \(2x^2\) (the term with \(x^2\))
  • Linear term: \(-3x\) (the term with \(x\))
  • Constant term: \(-3\) (the term with no \(x\))

Thus, the types of terms that appear in the polynomial expression are:

  • Constant
  • Linear
  • Quadratic

The expression does not contain a cubic term. Therefore, the correct types of terms are Constant, Linear, and Quadratic.

Use the Distributive Property to rewrite the expression and then simplify.
2(x2−x)+6x−3−7x
Responses:
−x−3.
2x2−x−3.
2x2−3x−3.
−4x.

Let's use the Distributive Property to rewrite the expression \(2(x^2 - x) + 6x - 3 - 7x\) step by step and then simplify.

  1. Start by applying the Distributive Property to the term \(2(x^2 - x)\): \[ 2(x^2 - x) = 2x^2 - 2x \]

  2. Now, substitute this back into the expression: \[ 2x^2 - 2x + 6x - 3 - 7x \]

  3. Next, combine like terms. For the \(x\) terms:

    • Combine \(-2x\), \(6x\), and \(-7x\): \[ -2x + 6x - 7x = (-2 + 6 - 7)x = -3x \]

  4. Now, write the expression with the combined terms: \[ 2x^2 - 3x - 3 \]

Thus, the simplified expression is: \[ 2x^2 - 3x - 3 \]

From the provided responses, the correct answer is \(2x^2 - 3x - 3\).

Use the Commutative Property to rewrite the expression in a different order.
2x2−3x−3
Responses:
−3−3x+2x2
-3-x
3−2x2+x
3−3x−2x2

The Commutative Property allows us to change the order of the terms in an expression without changing the result.

Given the expression \(2x^2 - 3x - 3\), we can rewrite it in a different order. Here are the terms in the expression: \(2x^2\), \(-3x\), and \(-3\).

Let's rearrange these terms using the Commutative Property:

  1. We can write it as: \[ -3 + (-3x) + (2x^2) \] or simply: \[ -3 - 3x + 2x^2 \]

This matches one of the response options given, which is: \(-3 - 3x + 2x^2\).

This would be the correct answer based on the use of the Commutative Property.

Johnny rewrites the expression, 2x2−3x−3, as the expression shown below. Explain why this statement is or is not equivalent to the original expression.
2x2−(3x+3)
Responses:
No, they are NOT equivalent because if you distribute the negative sign, the last number will be positive 3 and in the original expression it is -3.

No, they are NOT equivalent because if you simplify that expression, you get -6x at the very end and that is not in the original expression.

Yes, they are equivalent because if you simplify the expression, you get a 6x at the end and the original expression has a 6x.

Yes, they are equivalent because if you distribute the negative sign, you get -3x and -3 and that is the same in the simplified original expression .

To determine whether the expression \(2x^2 - 3x - 3\) is equivalent to the expression \(2x^2 - (3x + 3)\), let's analyze the second expression by distributing the negative sign:

Starting with \(2x^2 - (3x + 3)\):

  1. Distribute the negative sign: \[ 2x^2 - 3x - 3 \]

Now, we can compare this result with the original expression:

  • Original Expression: \(2x^2 - 3x - 3\)
  • Rewritten Expression: \(2x^2 - 3x - 3\) (after distribution)

Since both expressions simplify to the same terms, they are indeed equivalent.

Therefore, the correct response is:

Yes, they are equivalent because if you distribute the negative sign, you get -3x and -3 and that is the same in the simplified original expression.

Pick the expression for that has the same number of terms as the following expression:
2x2−3x−3
Responses:
7x2+3
6x2−9x−8
4x
−4

The original expression \(2x^2 - 3x - 3\) contains 3 terms:

  1. \(2x^2\) (quadratic term)
  2. \(-3x\) (linear term)
  3. \(-3\) (constant term)

Now, let's examine the provided response options to find which has the same number of terms (3 terms):

  1. \(7x^2 + 3\) - This has 2 terms.
  2. \(6x^2 - 9x - 8\) - This has 3 terms.
  3. \(4\) - This has 1 term.
  4. \(-4\) - This has 1 term.

The only expression that has the same number of terms (3 terms) as the original expression \(2x^2 - 3x - 3\) is:

\(6x^2 - 9x - 8\).

dd the following polynomials:
(2x2−3x−3 )+(6x2−9x−8)
Responses: -4x^2-12x-11
8x2−12x−11
8x2+6x−5
−4x2− 12x+11

To add the polynomials \((2x^2 - 3x - 3) + (6x^2 - 9x - 8)\), we follow these steps:

  1. Combine like terms:
    • For the \(x^2\) terms: \(2x^2 + 6x^2 = 8x^2\)
    • For the \(x\) terms: \(-3x - 9x = -12x\)
    • For the constant terms: \(-3 - 8 = -11\)

Putting it all together, we get: \[ 8x^2 - 12x - 11 \]

Now, looking at the response options provided, the correct answer is:

8x² - 12x - 11.