Multiply a polynomial by a monomial

Multiplying a polynomial by a monomial means applying the distributive property to every term, multiplying numerical coefficients and combining powers of like variables by adding exponents. The result is an equivalent polynomial, with attention to signs, zero coefficients, and the distinction between multiplying powers and adding them; this understanding supports expanding expressions, simplifying algebraic forms, and later factoring.

Detailed Explanation: 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.

Learn by doing: Multiply a polynomial by a monomial

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Radicals - Multiplying Monomials with Binomials (Values Only)

Earned ?