Question

1. find each of the following products.

(a) $a^3(a + a^2)$
(b) $b^2(4 - b)$
(c) $-3c(4c + c^2)$
(d) $\frac{d^2}{2}(-1 + 2d)$

Explanation:

Step1: Distribute $a^3$ to terms inside

$a^3(a + a^2) = a^3 \cdot a + a^3 \cdot a^2$

Step2: Apply exponent rule $x^m \cdot x^n=x^{m+n}$

$a^{3+1} + a^{3+2} = a^4 + a^5$

Step1: Distribute $b^2$ to terms inside

$b^2(4 - b) = b^2 \cdot 4 - b^2 \cdot b$

Step2: Apply exponent rule $x^m \cdot x^n=x^{m+n}$

$4b^2 - b^{2+1} = 4b^2 - b^3$

Step1: Distribute $-3c$ to terms inside

$-3c(4c + c^2) = -3c \cdot 4c + (-3c) \cdot c^2$

Step2: Calculate products and exponents

$-12c^{1+1} - 3c^{1+2} = -12c^2 - 3c^3$

Step1: Distribute $\frac{d^2}{2}$ to terms inside

$\frac{d^2}{2}(-1 + 2d) = \frac{d^2}{2} \cdot (-1) + \frac{d^2}{2} \cdot 2d$

Step2: Simplify each term

$-\frac{d^2}{2} + \frac{2d^{2+1}}{2} = -\frac{d^2}{2} + d^3$

Answer:

(a) $a^4 + a^5$
(b) $4b^2 - b^3$
(c) $-12c^2 - 3c^3$
(d) $-\frac{d^2}{2} + d^3$