Problem 1: Find the degree of the monomial $5x^3y^2$.
Problem 2: Find the degree of the monomial $-7a^2b^4c$.
Problem 3: What is the degree of the constant term $12$?
Problem 4: State whether the degree of $0$ is defined or undefined.
Problem 5: Find the degree of the polynomial $4x^2y + 3xy^3 – 5x + 6$.
Problem 6: Find the degree of the polynomial $7a^3 – 2a^2b + ab^2 + 9$.
Problem 7: Find the degree of each term in the polynomial $6x^4y^2 – 3x^2y + y$.
Problem 8: Using substitution, find the value of $3x – 5$ when $x = 4$.
Problem 9: Find the value of $2a^2b$ when $a = -2$ and $b = 3$.
Problem 10: Evaluate the expression $x^2 + 2x – 1$ when $x = -1$.
1. Degree of $5x^3y^2$
Answer: 5
Explanation: To find the degree of a monomial, you add the exponents of all the variables. Here, the exponent of $x$ is 3 and the exponent of $y$ is 2. Therefore, $3 + 2 = 5$.
2. Degree of $-7a^2b^4c$
Answer: 7
Explanation: Add the exponents: $a$ has 2, $b$ has 4, and $c$ has an “invisible” exponent of 1. Summing them up ($2 + 4 + 1$) gives a degree of 7.
3. Degree of the constant term $12$
Answer: 0
Explanation: A constant (a number without a variable) can be thought of as being multiplied by a variable to the power of 0 (e.g., $12x^0$). Since $x^0 = 1$, the degree is always 0.
4. Degree of $0$
Answer: Undefined
Explanation: While non-zero constants have a degree of 0, the number 0 itself is a special case. Because $0$ times any power of $x$ is still 0, there is no unique exponent that can be assigned to it; therefore, the degree is undefined.
5. Degree of $4x^2y + 3xy^3 – 5x + 6$
Answer: 4
Explanation: The degree of a polynomial is the highest degree of any of its individual terms. The term $4x^2y$ has degree 3, and $3xy^3$ has degree 4. Since 4 is the highest, that is the degree of the whole polynomial.
6. Degree of $7a^3 – 2a^2b + ab^2 + 9$
Answer: 3
Explanation: Examining each term: $7a^3$ is degree 3, $-2a^2b$ is degree 3 ($2+1$), and $ab^2$ is also degree 3 ($1+2$). The highest degree present is 3.
7. Degree of each term in $6x^4y^2 – 3x^2y + y$
Answer: 6, 3, and 1
Explanation: For the first term ($6x^4y^2$), $4+2=6$. For the second term ($-3x^2y$), $2+1=3$. For the last term ($y$), the exponent is 1.
8. Value of $3x – 5$ when $x = 4$
Answer: 7
Explanation: Replace the $x$ with 4: $3(4) – 5$. Following the order of operations, multiply first to get $12 – 5$, which equals 7.
9. Value of $2a^2b$ when $a = -2$ and $b = 3$
Answer: 24
Explanation: Substitute the values: $2(-2)^2(3)$. First, square the $-2$ to get 4. Then multiply: $2 \times 4 \times 3 = 24$.
10. Value of $x^2 + 2x – 1$ when $x = -1$
Answer: -2
Explanation: Substitute $-1$ for $x$: $(-1)^2 + 2(-1) – 1$. This simplifies to $1 – 2 – 1$. Subtracting across gives $1 – 2 = -1$, and $-1 – 1 = -2$.