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Multiply a monomial by a polynomial

Multiplying a monomial by a polynomial means applying the distributive property to multiply the monomial by every term, multiplying numerical coefficients and adding exponents of matching variables according to the laws of exponents. The result is written as a polynomial in simplified form, with like terms combined when they arise; the scope includes integer or rational coefficients and nonnegative integer exponents, but excludes noninteger or negative exponents and abstract polynomial structures, supporting later expansion, factoring, and equation solving.

Detailed Explanation: Multiply a monomial by a polynomial

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

When multiplying:

  • Multiply the numerical coefficients.
  • For matching variables, add the exponents:
xmâ‹…xn=xm+nx^m\cdot x^n=x^{m+n}
  • Write the answer in descending order of exponents.

Consider the example:

3x2(2x3−5x+4)3x^2(2x^3-5x+4)

Step 1: Distribute 3x23x^2 to each term.

3x2(2x3)−3x2(5x)+3x2(4)3x^2(2x^3)-3x^2(5x)+3x^2(4)

Step 2: Multiply each term.

For the first term:

3x2â‹…2x3=6x2+3=6x53x^2\cdot 2x^3=6x^{2+3}=6x^5

For the second term:

3x2⋅(−5x)=−15x2+1=−15x33x^2\cdot(-5x)=-15x^{2+1}=-15x^3

For the third term:

3x2â‹…4=12x23x^2\cdot 4=12x^2

Step 3: Write the simplified polynomial.

6x5−15x3+12x2\boxed{6x^5-15x^3+12x^2}

Always check that the monomial was multiplied by every term inside the parentheses.

Learn by doing: Multiply a monomial by a polynomial

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    Polynomial Multiply - Monomial by Trinomial, Order 2, Two Variables

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