examine the following expression.
p² - 3 + 3p - 8 + p + p³
which statements about the expression are true? check all that apply.
□ the constants, −3 and −8, are like terms.
□ the terms 3p and p are like terms.
□ the terms in the expression are p², −3, 3p, −8, p, p³.
□ the terms p², 3p, p, and p³ have variables, so they are like terms.
□ the expression contains six terms.
□ the terms p² and p³ are like terms.
□ like terms have the same variables raised to the same powers.

Question

Accepted Answer

# Explanation:
## Step1: Recall like terms definition
Like terms have same variable(s) raised to same power. Constants (terms without variables) are like terms.
## Step2: Analyze each statement
- "The constants, -3 and -8, are like terms": Constants are like terms (no variables, same "power" of 0). True.
- "The terms $3p$ and $p$ are like terms": Both have variable $p$ raised to power 1. True.
- "The terms in the expression are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$": Let's simplify the expression $p^2 - 3 + 3p - 8 + p + p^3$. Combine like terms (constants: $-3-8=-11$; $p$ terms: $3p + p = 4p$). The terms are $p^3$, $p^2$, $4p$, $-11$? Wait, no—original expression's terms (before combining) are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$. So 6 terms? Wait, no: $p^2$ (1), $-3$ (2), $3p$ (3), $-8$ (4), $p$ (5), $p^3$ (6). Wait, but when we list terms, we consider each part separated by + or -. So yes, the terms are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$. So this statement is true? Wait, no—wait the expression is $p^2 - 3 + 3p - 8 + p + p^3$. So the terms are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$ (6 terms). Wait, but let's check other statements.
- "The terms $p^2$, $3p$, $p$, and $p^3$ have variables, so they are like terms": $p^2$ (power 2), $3p$ (power 1), $p$ (power 1), $p^3$ (power 3). Different powers, so not like terms. False.
- "The expression contains six terms": Original expression: $p^2 - 3 + 3p - 8 + p + p^3$. Terms: $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$ → 6 terms. Wait, but when we write the expression, each term is separated by + or -. So yes, 6 terms? Wait, but let's see: $p^2$ (1), $-3$ (2), $3p$ (3), $-8$ (4), $p$ (5), $p^3$ (6). So this statement is true? Wait, but let's check the next statement.
- "The terms $p^2$ and $p^3$ are like terms": $p^2$ (power 2), $p^3$ (power 3) → different powers. False.
- "Like terms have the same variables raised to the same powers": This is the definition of like terms. True.

Wait, let's re-express the original expression: $p^3 + p^2 + 3p + p - 3 - 8$ (combining $3p$ and $p$ as like terms, $-3$ and $-8$ as like terms). Wait, no—before combining, the terms are $p^3$, $p^2$, $3p$, $p$, $-3$, $-8$ (6 terms). So:

1. "The constants, -3 and -8, are like terms": True (both constants).
2. "The terms $3p$ and $p$ are like terms": True (both $p^1$).
3. "The terms in the expression are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$": True (these are the terms before combining like terms).
4. "The terms $p^2$, $3p$, $p$, and $p^3$ have variables, so they are like terms": False (different powers).
5. "The expression contains six terms": True (6 terms: $p^3$, $p^2$, $3p$, $p$, $-3$, $-8$ – wait, original expression is $p^2 - 3 + 3p - 8 + p + p^3$, so terms are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$ → 6 terms. True? Wait, but when we count terms, each signed part is a term. So yes, 6 terms.
6. "The terms $p^2$ and $p^3$ are like terms": False (different exponents).
7. "Like terms have the same variables raised to the same powers": True (definition).

Wait, but the options given are:

1. The constants, -3 and -8, are like terms. (True)
2. The terms $3p$ and $p$ are like terms. (True)
3. The terms in the expression are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$. (True, as these are the terms before combining)
4. The terms $p^2$, $3p$, $p$, and $p^3$ have variables, so they are like terms. (False)
5. The expression contains six terms. (True, as there are six terms before combining)
6. The terms $p^2$ and $p^3$ are like terms. (False)
7. Like terms have the same variables raised to the same powers. (True)

But let's check the original problem's options (from the image):

Options (checkboxes):

- The constants, -3 and -8, are like terms. (True)
- The terms $3p$ and $p$ are like terms. (True)
- The terms in the expression are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$. (True)
- The terms $p^2$, $3p$, $p$, and $p^3$ have variables, so they are like terms. (False)
- The expression contains six terms. (True, since the terms are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$ → 6 terms)
- The terms $p^2$ and $p^3$ are like terms. (False)
- Like terms have the same variables raised to the same powers. (True)

But maybe the problem expects us to check which are true. Let's confirm each:

1. Constants -3 and -8: like terms (yes, constants are like terms). True.
2. $3p$ and $p$: like terms (same variable $p$, same exponent 1). True.
3. Terms are $p^2$, -3, 3p, -8, p, p^3: Let's expand the expression $p^2 - 3 + 3p - 8 + p + p^3$. The terms (separated by + or -) are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$. So 6 terms. True.
4. $p^2$, 3p, p, p^3: like terms? No, different exponents (2,1,1,3). False.
5. Expression has six terms: As above, yes (6 terms). True.
6. $p^2$ and $p^3$: like terms? No, exponents 2 vs 3. False.
7. Like terms definition: True.

But maybe the problem's options are as listed, and we need to check the true ones. Let's re-express the original expression's terms:

Original expression: $p^3 + p^2 + 3p + p - 3 - 8$ (after reordering). Wait, no—original is $p^2 - 3 + 3p - 8 + p + p^3$, so the terms (before combining) are:

1. $p^2$
2. $-3$
3. $3p$
4. $-8$
5. $p$
6. $p^3$

So 6 terms. So "The expression contains six terms" is true.

"Like terms have the same variables raised to the same powers" is the definition, so true.

"The constants, -3 and -8, are like terms" – true (both constants).

"The terms $3p$ and $p$ are like terms" – true (both $p^1$).

"The terms in the expression are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$" – true (these are the terms before combining).

The other two are false.

# Answer:
The true statements are:
- The constants, -3 and -8, are like terms.
- The terms $3p$ and $p$ are like terms.
- The terms in the expression are $p^2$, $-3$, $3p$, $-8$, $p$, $p^3$.
- The expression contains six terms.
- Like terms have the same variables raised to the same powers.

(Note: Depending on the problem's exact options, but based on analysis, these are true.)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall like terms definition

Step2: Analyze each statement

Answer: