Question

question 5 of 5
select the correct locations on the image.
jennifer is rewriting a polynomial by combining like terms. which terms does she still need to combine to finish rewriting the polynomial?
$4cd^2 - 7cd^2 + 3d^2 + cd^2 - 4cd^2 - c^2$

Explanation:

Step1: Identify like terms

Like terms have the same variables raised to the same powers. Here, the terms with \( cd^2 \) are \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \). The term \( 3d^2 \) has only \( d^2 \) (no \( c \)), and \( -c^2 \) has only \( c^2 \) (no \( d \)). So we need to combine the \( cd^2 \) terms and check if any others are left, but the question is about which terms are left to combine. Wait, let's list all \( cd^2 \) terms: \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \), and then \( 3d^2 \) and \( -c^2 \). Wait, maybe I misread. Wait the polynomial is \( 4cd^2 - 7cd^2 + 3d^2 + cd^2 - 4cd^2 - c^2 \). So first, combine the \( cd^2 \) terms. Let's calculate their coefficients: \( 4 - 7 + 1 - 4 \). Let's do that: \( 4 -7 = -3 \), \( -3 +1 = -2 \), \( -2 -4 = -6 \). Wait, no, wait the terms are \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \). So combining those: \( (4 - 7 + 1 - 4)cd^2 = (4 +1 -7 -4)cd^2 = (5 - 11)cd^2 = -6cd^2 \). Then the remaining terms are \( 3d^2 \) and \( -c^2 \), which are not like terms. But the question is "Which terms does she still need to combine to finish rewriting the polynomial?" Wait, maybe the like terms are the \( cd^2 \) terms? Wait no, maybe I made a mistake. Wait the original polynomial: \( 4cd^2 - 7cd^2 + 3d^2 + cd^2 - 4cd^2 - c^2 \). So the \( cd^2 \) terms are \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \). Let's combine them: \( 4 -7 +1 -4 = (4 +1) + (-7 -4) = 5 -11 = -6 \), so \( -6cd^2 \). Then the other terms are \( 3d^2 \) and \( -c^2 \), which are not like terms. Wait, but maybe the question is that after combining some, which are left? Wait, maybe the \( cd^2 \) terms are not fully combined? Wait no, we combined all \( cd^2 \) terms. Wait, maybe the problem is that the \( 3d^2 \) and \( -c^2 \) are not like terms, but the \( cd^2 \) terms were combined? Wait, no, maybe I misread the polynomial. Wait the user's image shows the polynomial as \( 4cd^2 - 7cd^2 + 3d^2 + cd^2 - 4cd^2 - c^2 \). So the like terms for \( cd^2 \) are all four terms: \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \). So she needs to combine those \( cd^2 \) terms (since they are like terms) and then the remaining terms \( 3d^2 \) and \( -c^2 \) are not like terms, so they stay. Wait, but the question is which terms are left to combine. So the \( cd^2 \) terms are like terms and need to be combined. Wait, maybe the answer is the \( cd^2 \) terms? Wait, no, let's re-express. The polynomial has terms: \( 4cd^2 \), \( -7cd^2 \), \( 3d^2 \), \( cd^2 \), \( -4cd^2 \), \( -c^2 \). So the like terms are the four with \( cd^2 \). So she still needs to combine \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \) (the \( cd^2 \) terms) and then the remaining are \( 3d^2 \) and \( -c^2 \), which can't be combined. Wait, maybe the question is that after combining some, which are left. Wait, maybe the initial combination was partial. Wait, perhaps the user's question is about identifying the like terms that are left to combine. So the terms with \( cd^2 \) are like terms and need to be combined. So the terms \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \) are like terms and need to be combined. Alternatively, maybe the \( 3d^2 \) and \( -c^2 \) are not like terms, but the \( cd^2 \) terms are the ones to combine. Wait, maybe I made a mistake in the problem understanding. Let's re-express the polynomial:

\( 4cd^2 - 7cd^2 + 3d^2 + cd^2 - 4cd^2 - c^2 \)

Group like terms:

\( (4cd^2 - 7cd^2 + cd^2 - 4cd^2) + 3d^2 - c^2 \)

Now, combine the \( cd^2 \) terms:

\( 4 - 7 + 1 - 4 = (4 + 1) - (7 + 4) = 5 - 11 = -6 \)

So the combined polynomial is \( -6cd^2 + 3d^2 - c^2 \). But the question is "Which terms does she still need to combine to finish rewriting the polynomial?" Wait, maybe the answer is the \( cd^2 \) terms (i.e., \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \)) because they are like terms and need to be combined. Alternatively, maybe the \( 3d^2 \) and \( -c^2 \) are not like terms, so they can't be combined, but the \( cd^2 \) terms are the ones left to combine. Wait, perhaps the question is that after combining some, the remaining like terms? But \( 3d^2 \) and \( -c^2 \) are not like terms. So the only like terms left are the \( cd^2 \) terms. So she needs to combine \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \).

Step2: Confirm

So the terms with \( cd^2 \) are like terms and need to be combined. So the terms she still needs to combine are \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \) (the \( cd^2 \) terms) or maybe the remaining after partial combination? Wait, maybe the problem is that the user's image has a selection, but since we can't see the image, we have to infer. But based on the polynomial, the like terms are the \( cd^2 \) terms. So the answer is the terms with \( cd^2 \): \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), \( -4cd^2 \).

Answer:

The terms \( 4cd^2 \), \( -7cd^2 \), \( cd^2 \), and \( -4cd^2 \) (the \( cd^2 \)-like terms) still need to be combined.