Multiply a Monomial and a Polynomial

How to multiply a monomial and a polynomial: 2 examples and their solutions.

Example 1

Example

Solution

Multiply x and each term of (3x2 - 5x + 8).

First multiply x and 3x2: x⋅3x2.

Multiply x and -5x: + x⋅(-5x).

Multiply x and +8: + x⋅8.

So x(3x2 - 5x + 8) = x⋅3x2 + x⋅(-5x) + x⋅8.

x⋅3x2 = 3x3
+x⋅(-5x) = -5x2
+x⋅8 = +8x

Product of Powers

So 3x3 - 5x2 + 8x is the answer.

Example 2

Example

Solution

Multiply each term of (5 - a + 3b) and a.

First multiply 5 and a: 5⋅a.

Multiply -a and a: -a⋅a.

Multiply +3b and a: +3b⋅a.

Next, multiply -3 (not 3) and each term of (a2 + ab - 2).

Multiply -3 and a2: -3⋅a2.

Multiply -3 and +ab: -3⋅ab.

Multiply -3 and -2: -3⋅(-2).

So the given expression is
5⋅a - a⋅a + 3b⋅a - 3⋅a2 - 3⋅ab - 3⋅(-2).

5⋅a = 5a
- a⋅a = -a2
+3b⋅a = +3ab

-3⋅a2 = -3a2
-3⋅ab = -3ab
-3⋅(-2) = +6

-a2 and -3a2 are like terms.
So -a2 - 3a2 = (-1 - 3)a2 = -4a2.

Cancel +3ab and -3ab.
You can cancel these two terms because
+3ab - 3ab = (3 - 3)ab = 0⋅ab = 0.

Add and Subtract Polynomials

So -4a2 + 5a + 6 is the answer.