Solution: To simplify this we must multiply the binomial by itself using the FOIL method.

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1 Special Products This section of notes will focus on the use of formulas to find products. Although it may seem like a lot of extra memorizing, these formulas will save considerable time when multiplying special products. The Square of a Binomial: To square a binomial means to raise it to the nd power or to multiply it by itself. To accomplish this we may use the FOIL method and multiply as previously learned. ( x +. Solution: To simplify this we must multiply the binomial by itself using the FOIL method. ( x + = ( x + ( x + + x + x x + 9 Therefore, ( x + + 6x + 9. Notice that the resulting trinomial is the square of the first term plus twice the product of the two terms plus the square of the last term in the binomial. This will always be the case and leads to a formula for squaring any binomial ( a + b = a + ab + b Although the FOIL method may always be used, memorizing this formula will save a tremendous amount of time squaring binomials throughout this course. ( x + 6 Solution: In this example a and b = 6. Applying the formula we have, ( x + 6 = ( x + ( x(6 + (6 + x + 6 ( q 7 Solution: In this example a = q and b = -7. Applying the formula we have, ( q 7 = ( q = q + ( q( 7 + ( 7 q + 9

2 Once we know and understand the formula, we may do the st step mentally further simplifying the process. ( x + 9 Solution: In this example a and b = 9. Applying the formula we have, ( x x + 8 ( x y Solution: In this example a = x and b = -y. Applying the formula we have, (x y = x 0xy + 6y ( r + s Solution: In this example a = r and b = s. Applying the formula we have, (r = r + s + r s + 9s 6 ( x 7 y Solution: In this example a = x and b = 7 y. Applying the formula we have, (x = 9x 7y 0 x y + 9y 8 Multiplying Conjugates: Two binomials that are identical except for the sign between the two terms are conjugates. Together they make up a conjugate pair. In general, a conjugate pair is of the form: ( a + b( a b As always, we may use the FOIL method to multiply conjugates.

3 Example: Multiply ( x + ( x. Solution: First, notes that these binomials are conjugates. ( x + ( x x + x 6 6 Therefore, ( x + ( x 6. Notice that when multiplying conjugates the middle terms will cancel out leaving only the product of the st terms plus the product of the nd terms. This again will always be the case and leads to the formula: ( a + b( a b = a b Although the FOIL method may always be used to multiply conjugates, memorizing this simple formula will save a tremendous amount of time. Example: Multiply ( x + ( x Solution: In this example a and b =. Applying the formula we have, ( x + ( x Example: Multiply ( x + y( x y Solution: In this example a and b = y. Applying the formula we have, ( x + y( x y y Example: Multiply ( x z(x + z Solution: In this example a = x and b = z. Applying the formula we have, (x z(x + z = 9x 6z

4 Example: Multiply ( p q( p + q Solution: In this example a = p and b = q. Applying the formula we have, ( p q( p + q 6 p q Example: Multiply x x + Solution: In this example a and b =. Applying the formula we have, x x + x 9

5 Applications: Example: In a blood vessel of radius r at point b units from the center of the blood vessel, the blood flow speed is given by the expression k( r + b( r b, where k is a constant. Write this expression without parentheses. Solution: To write this expression without parentheses, we need to multiply. k( r + b( r b = k( r = kr b kb Example: The area of a square wooden frame can be represented by the polynomial ( S + s( S s where s is the side of the smaller square and S is the side of the larger square. Write this expression without parentheses. Solution: To write this expression without parentheses, we need to multiply. ( S + s( S s = S s Example: A student deposits $,000 in an account at an interest rate r. If the interest earned is compounded annually, the amount in the account after years is given by A = 000( + r. Write A as a polynomial in terms of r without parentheses. Solution: Once again, we need to multiply. A = 000( + r A = 000( + r + r A = r + 000r Example: As the temperature of a light bulb s filament changes from T tot, the energy that the filament radiates changes by the quantity a ( T T ( T + T ( T +. Multiply to find this change. T Solution: a( T T = a( T = a( T = at ( T T T at ( T + T ( T + T + T