Question

bellringer
multiply the following
polynomials

1. $(x^3 + 4x^2 + 2)(x^2 + 3x)$
2. $(x^2 + 4x)(-x^3 + 5)$

Explanation:

Problem 1: \((x^3 + 4x^2 + 2)(x^2 + 3x)\)

Step 1: Distribute each term

Use the distributive property (FOIL for polynomials) to multiply each term in the first polynomial by each term in the second polynomial.
\[

$$\begin{align*} &x^3 \cdot x^2 + x^3 \cdot 3x + 4x^2 \cdot x^2 + 4x^2 \cdot 3x + 2 \cdot x^2 + 2 \cdot 3x\\ \end{align*}$$

\]

Step 2: Simplify each product

Simplify each term using the rule of exponents \(a^m \cdot a^n = a^{m + n}\).
\[

$$\begin{align*} &x^{3+2} + 3x^{3+1} + 4x^{2+2} + 12x^{2+1} + 2x^2 + 6x\\ &= x^5 + 3x^4 + 4x^4 + 12x^3 + 2x^2 + 6x\\ \end{align*}$$

\]

Step 3: Combine like terms

Combine the terms with the same exponent.
\[

$$\begin{align*} &x^5 + (3x^4 + 4x^4) + 12x^3 + 2x^2 + 6x\\ &= x^5 + 7x^4 + 12x^3 + 2x^2 + 6x\\ \end{align*}$$

\]

Problem 2: \((x^2 + 4x)(-x^3 + 5)\)

Step 1: Distribute each term

Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial.
\[

$$\begin{align*} &x^2 \cdot (-x^3) + x^2 \cdot 5 + 4x \cdot (-x^3) + 4x \cdot 5\\ \end{align*}$$

\]

Step 2: Simplify each product

Simplify each term using the rule of exponents \(a^m \cdot a^n = a^{m + n}\).
\[

$$\begin{align*} &-x^{2+3} + 5x^2 - 4x^{1+3} + 20x\\ &= -x^5 + 5x^2 - 4x^4 + 20x\\ \end{align*}$$

\]

Step 3: Rearrange terms

Rearrange the terms in descending order of exponents.
\[

$$\begin{align*} &-x^5 - 4x^4 + 5x^2 + 20x\\ \end{align*}$$

\]

Answer:

s:

1. \(x^5 + 7x^4 + 12x^3 + 2x^2 + 6x\)
2. \(-x^5 - 4x^4 + 5x^2 + 20x\)