1. **Stating the problem:** We have two polynomials: $$P = 4x^3 + 3x^2 - x + 3$$ $$Q = x^3 - x + 1$$ We need to find: a) The degree of $P+Q$. b) The degree of $PQ$. c) The coefficient of the $x^3$ term in $PQ$. 2. **Find the degree of $P+Q$:** Add $P$ and $Q$ term by term: $$P+Q = (4x^3 + 3x^2 - x + 3) + (x^3 - x + 1)$$ Combine like terms: $$= (4x^3 + x^3) + 3x^2 + (-x - x) + (3 + 1)$$ $$= 5x^3 + 3x^2 - 2x + 4$$ The highest power of $x$ with a nonzero coefficient is $3$, so the degree of $P+Q$ is $3$. 3. **Find the degree of $PQ$:** The degree of a product of polynomials is the sum of their degrees. Degree of $P$ is $3$ (highest power in $4x^3$). Degree of $Q$ is $3$ (highest power in $x^3$). Therefore, $$\text{degree}(PQ) = 3 + 3 = 6$$ 4. **Find the coefficient of the $x^3$ term in $PQ$:** Write $P$ and $Q$ explicitly: $$P = 4x^3 + 3x^2 - x + 3$$ $$Q = x^3 + 0x^2 - x + 1$$ To find the coefficient of $x^3$ in $PQ$, consider all pairs of terms from $P$ and $Q$ whose powers add to $3$: - $4x^3$ from $P$ and constant term $1$ from $Q$: power sum $3 + 0 = 3$, product $4 \times 1 = 4$ - $3x^2$ from $P$ and $-x$ from $Q$: power sum $2 + 1 = 3$, product $3 \times (-1) = -3$ - $-x$ from $P$ and $0x^2$ from $Q$: power sum $1 + 2 = 3$, product $-1 \times 0 = 0$ - $3$ from $P$ and $x^3$ from $Q$: power sum $0 + 3 = 3$, product $3 \times 1 = 3$ Sum these contributions: $$4 + (-3) + 0 + 3 = 4$$ **Final answers:** a) Degree of $P+Q$ is $3$. b) Degree of $PQ$ is $6$. c) Coefficient of $x^3$ in $PQ$ is $4$.