1. Stating the problem: Given the polynomial $$P(x) = (x + 2)^2 (x + 3)^3 (x - 1)^4 (2x + 1)$$ we need to find its standard form, leading term, leading coefficient, degree, constant term, y-intercept, x-intercepts, and a table of values. 2. Expand each factor partially to find highest degree term for leading term and degree: - Highest degree from $(x + 2)^2$ is $x^2$ - Highest degree from $(x + 3)^3$ is $x^3$ - Highest degree from $(x - 1)^4$ is $x^4$ - Highest degree from $(2x + 1)$ is $2x$ Multiplying these degrees: $x^2 \cdot x^3 \cdot x^4 \cdot x = x^{2+3+4+1} = x^{10}$ 3. Find leading coefficient by multiplying leading coefficients of each factor’s highest degree term: - Coefficient from $(x + 2)^2$ highest term: $1$ - Coefficient from $(x + 3)^3$ highest term: $1$ - Coefficient from $(x - 1)^4$ highest term: $1$ - Coefficient from $(2x + 1)$ highest term: $2$ Multiplying these: $1 \times 1 \times 1 \times 2 = 2$ Leading term: $$2x^{10}$$ Leading coefficient: $$2$$ Degree: $$10$$ 4. Find the constant term by substituting $x=0$ into the fully factored form: $$P(0)=(0+2)^2 (0+3)^3 (0-1)^4 (2\cdot0 +1) = 2^2 \times 3^3 \times (-1)^4 \times 1 = 4 \times 27 \times 1 \times 1=108$$ Constant term and y-intercept (value when $x=0$): $$108$$ 5. X-intercepts are values of $x$ where $P(x)=0$, directly from factors: - From $(x + 2)^2 = 0 \implies x = -2$ - From $(x + 3)^3=0 \implies x = -3$ - From $(x - 1)^4=0 \implies x = 1$ - From $(2x + 1) = 0 \implies x = -\frac{1}{2}$ 6. The polynomial in standard expanded form is very large to write fully here but can be expressed as the product: $$P(x) = (x + 2)^2 (x + 3)^3 (x - 1)^4 (2x + 1)$$ 7. Table of values for selected $x$: | x | P(x) | |---|-------| | -3 | 0 | | -2 | 0 | | -0.5 | 0 | | 0 | 108 | | 1 | 0 | | 2 | Calculate $P(2)$: $$P(2) = (2 + 2)^2 (2 + 3)^3 (2 - 1)^4 (2\cdot2 + 1) = 4^2 \times 5^3 \times 1^4 \times 5 = 16 \times 125 \times 1 \times 5 = 10000$$ | 3 | Calculate $P(3)$: $$P(3) = (3 + 2)^2 (3 + 3)^3 (3 - 1)^4 (2\cdot3 + 1) = 5^2 \times 6^3 \times 2^4 \times 7 = 25 \times 216 \times 16 \times 7 = 604800$$