Skill: Multiply a polynomial by a monomial

To multiply a polynomial by a monomial, use the distributive property: multiply the monomial by every term in the polynomial.

When multiplying powers with the same variable, multiply the coefficients and add the exponents:

$$a^m \cdot a^n=a^{m+n}$$

Example

Simplify:

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

Step 1: Distribute $4x^2$ to each term.

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

Step 2: Multiply each term.

$$4x^2\cdot 3x^3=12x^{2+3}=12x^5$$

$$4x^2\cdot(-2x)=-8x^{2+1}=-8x^3$$

$$4x^2\cdot 5=20x^2$$

Step 3: Write the resulting polynomial.

$$\boxed{12x^5-8x^3+20x^2}$$

Remember: exponents are added only when multiplying powers with the same base, such as $x^2\cdot x^3$. You do not add exponents across terms being added or subtracted.