Objectives Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2 + 3) 4y2(y2 + 3) 4y2  y2 + 4y2  3 Distribute. 4y4 + 12y2 Multiply. B. fg(f4 + 2f3g – 3f2g2 + fg3) fg(f4 + 2f3g – 3f2g2 + fg3) fg  f4 + fg  2f3g – fg  3f2g2 + fg  fg3 Distribute. f5g + 2f4g2 – 3f3g3 + f2g4 Multiply.

To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first. Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) Method 2 Multiply vertically. a2 – 5a + 2 a – 3 Write each polynomial in standard form. – 3a2 + 15a – 6 Multiply (a2 – 5a + 2) by –3. a3 – 5a2 + 2a Multiply (a2 – 5a + 2) by a, and align like terms. a3 – 8a2 + 17a – 6 Combine like terms.

Example 4: Expanding a Power of a Binomial Find the product. (a + 2b)3 (a + 2b)(a + 2b)(a + 2b) Write in expanded form. (a + 2b)(a2 + 4ab + 4b2) Multiply the last two binomial factors. Distribute a and then 2b. a(a2) + a(4ab) + a(4b2) + 2b(a2) + 2b(4ab) + 2b(4b2) a3 + 4a2b + 4ab2 + 2a2b + 8ab2 + 8b3 Multiply. a3 + 6a2b + 12ab2 + 8b3 Combine like terms.

Multiply the last two binomial factors. Check It Out! Example 4a Find the product. (x + 4)4 (x + 4)(x + 4)(x + 4)(x + 4) Write in expanded form. (x + 4)(x + 4)(x2 + 8x + 16) Multiply the last two binomial factors. (x2 + 8x + 16)(x2 + 8x + 16) Multiply the first two binomial factors. Distribute x2 and then 8x and then 16. x2(x2) + x2(8x) + x2(16) + 8x(x2) + 8x(8x) + 8x(16) + 16(x2) + 16(8x) + 16(16) Multiply. x4 + 8x3 + 16x2 + 8x3 + 64x2 + 128x + 16x2 + 128x + 256 x4 + 16x3 + 96x2 + 256x + 256 Combine like terms.

Notice the coefficients of the variables in the final product of (a + b)3. these coefficients are the numbers from the third row of Pascal's triangle. Each row of Pascal’s triangle gives the coefficients of the corresponding binomial expansion. The pattern in the table can be extended to apply to the expansion of any binomial of the form (a + b)n, where n is a whole number.

This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.

Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k – 5)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. [1(k)3(–5)0] + [3(k)2(–5)1] + [3(k)1(–5)2] + [1(k)0(–5)3] k3 – 15k2 + 75k – 125 B. (6m – 8)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. [1(6m)3(–8)0] + [3(6m)2(–8)1] + [3(6m)1(–8)2] + [1(6m)0(–8)3] 216m3 – 864m2 + 1152m – 512

Identify the coefficients for n = 4, or row 4. Check It Out! Example 5 Expand the expression. c. (3x + 1)4 1 4 6 4 1 Identify the coefficients for n = 4, or row 4. [1(3x)4(1)0] + [4(3x)3(1)1] + [6(3x)2(1)2] + [4(3x)1(1)3] + [1(3x)0(1)4] 81x4 + 108x3 + 54x2 + 12x + 1