Answer :
To determine which expression evaluates to 3 when \( p = -2 \) and \( q = 3 \), let's evaluate each of the given expressions step-by-step.
1. Expression 1: \( p^2 + q - 4 q^2 \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( q \):
[tex]\[ q = 3 \][/tex]
3. Calculate \( 4 q^2 \):
[tex]\[ 4 q^2 = 4 \times (3)^2 = 4 \times 9 = 36 \][/tex]
4. Evaluate the entire expression:
[tex]\[ p^2 + q - 4 q^2 = 4 + 3 - 36 = 7 - 36 = -29 \][/tex]
The result is \(-29\), not 3.
2. Expression 2: \( 3 p^2 + 2 p q - 6 q + 2 \)
1. Calculate \( 3 p^2 \):
[tex]\[ 3 p^2 = 3 \times (-2)^2 = 3 \times 4 = 12 \][/tex]
2. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = 2 \times -6 = -12 \][/tex]
3. Calculate \( -6 q \):
[tex]\[ -6 q = -6 \times 3 = -18 \][/tex]
4. Sum up with the constant term:
[tex]\[ 12 - 12 - 18 + 2 = 12 - 30 + 2 = -16 \][/tex]
The result is \(-16\), not 3.
3. Expression 3: \( p^3 + 2 p^2 q - p^2 + 2 p q + q \)
1. Calculate \( p^3 \):
[tex]\[ p^3 = (-2)^3 = -8 \][/tex]
2. Calculate \( 2 p^2 q \):
[tex]\[ 2 p^2 q = 2 \times (-2)^2 \times 3 = 2 \times 4 \times 3 = 24 \][/tex]
3. Calculate \( - p^2 \):
[tex]\[ - p^2 = -(-2)^2 = -4 \][/tex]
4. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = -12 \][/tex]
5. Sum up with \( q \):
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q = -8 + 24 - 4 - 12 + 3 = -8 + 24 - 4 - 12 + 3 = 3 \][/tex]
The result is \( 3 \), which matches the student's result.
4. Expression 4: \( p^2 + 3 q^2 - q^2 + p \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( 3 q^2 \):
[tex]\[ 3 q^2 = 3 \times (3)^2 = 3 \times 9 = 27 \][/tex]
3. Calculate \( - q^2 \):
[tex]\[ - q^2 = - (3)^2 = -9 \][/tex]
4. Evaluate \( p \):
[tex]\[ p = -2 \][/tex]
5. Sum up:
[tex]\[ p^2 + 3 q^2 - q^2 + p = 4 + 27 - 9 - 2 = 4 + 18 - 2 = 20 \][/tex]
The result is \( 20 \), not 3.
Therefore, the expression the student evaluated to get a result of 3 is:
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q \][/tex]
1. Expression 1: \( p^2 + q - 4 q^2 \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( q \):
[tex]\[ q = 3 \][/tex]
3. Calculate \( 4 q^2 \):
[tex]\[ 4 q^2 = 4 \times (3)^2 = 4 \times 9 = 36 \][/tex]
4. Evaluate the entire expression:
[tex]\[ p^2 + q - 4 q^2 = 4 + 3 - 36 = 7 - 36 = -29 \][/tex]
The result is \(-29\), not 3.
2. Expression 2: \( 3 p^2 + 2 p q - 6 q + 2 \)
1. Calculate \( 3 p^2 \):
[tex]\[ 3 p^2 = 3 \times (-2)^2 = 3 \times 4 = 12 \][/tex]
2. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = 2 \times -6 = -12 \][/tex]
3. Calculate \( -6 q \):
[tex]\[ -6 q = -6 \times 3 = -18 \][/tex]
4. Sum up with the constant term:
[tex]\[ 12 - 12 - 18 + 2 = 12 - 30 + 2 = -16 \][/tex]
The result is \(-16\), not 3.
3. Expression 3: \( p^3 + 2 p^2 q - p^2 + 2 p q + q \)
1. Calculate \( p^3 \):
[tex]\[ p^3 = (-2)^3 = -8 \][/tex]
2. Calculate \( 2 p^2 q \):
[tex]\[ 2 p^2 q = 2 \times (-2)^2 \times 3 = 2 \times 4 \times 3 = 24 \][/tex]
3. Calculate \( - p^2 \):
[tex]\[ - p^2 = -(-2)^2 = -4 \][/tex]
4. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = -12 \][/tex]
5. Sum up with \( q \):
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q = -8 + 24 - 4 - 12 + 3 = -8 + 24 - 4 - 12 + 3 = 3 \][/tex]
The result is \( 3 \), which matches the student's result.
4. Expression 4: \( p^2 + 3 q^2 - q^2 + p \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( 3 q^2 \):
[tex]\[ 3 q^2 = 3 \times (3)^2 = 3 \times 9 = 27 \][/tex]
3. Calculate \( - q^2 \):
[tex]\[ - q^2 = - (3)^2 = -9 \][/tex]
4. Evaluate \( p \):
[tex]\[ p = -2 \][/tex]
5. Sum up:
[tex]\[ p^2 + 3 q^2 - q^2 + p = 4 + 27 - 9 - 2 = 4 + 18 - 2 = 20 \][/tex]
The result is \( 20 \), not 3.
Therefore, the expression the student evaluated to get a result of 3 is:
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q \][/tex]
