## Multiplying Polynomials

The following are rules regarding the multiplying of variable expressions.

• Rule 1: To multiply monomials with the same base, keep the base and add the powers:
• x ax b= x a + b
• Rule 2: To raise a base to a power, keep the base and multiply the powers.
• ( x a) b= x ab
• Rule 3: To raise a product to a power, raise each factor in the product to that power.
• ( xy) a= x ay a

Example 1

Simplify each of the following multiplication problems and state which of the preceding rules was applied.

1. yy 5
2. ( x 4) 3
3. (–2 x 4 y 2 z 3) 5
4. a 3( a 2 b 3) 4
5. 6. 7. 8. To multiply monomials together, follow this procedure.

1. Multiply the numerical coefficients together.
2. Multiply the variables together.
3. Write the results as a product.

Example 2

Simplify each of the following.

1. (4 x 2)(3 x 3
2. (–8 a 3 b 2)(2 a 2 b 2) 3
3. (4 x 2)(3 x 3) = (4 × 3)( x 2 x 3) = 12 x 5
4. (–8 a 3 b 2)(2 a 2 b 2) 3 = (–8 a 3 b 2)(8 a 6 b 6) = –64 a 9 b 8

To multiply polynomials together, multiply each term in one polynomial by each term in the other polynomial. Then simplify if possible.

Example 3

Multiply each of the following.

1. 5 x(3 x 2 – 4 x + 2)
2. (4 x – 2)(3 x + 5)
3. ( x + y)( x 2xy + y 2

The following shows how each equation is multiplied both horizontally and vertically.

Equation (a) done horizontally: Equation (a) done vertically: Equation (b) done horizontally: Equation (b) done vertically: Equation (c) done horizontally: Equation (c) done vertically: Top
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