HW: Pg. 341-342 #13-61 eoo

Quiz 1 Pg. 344 #13-26

Vocabulary Polynomial Long Division: When you divide a polynomial ______ by a divisor _____, you get a quotient polynomial ______ and a remainder polynomial _____. This can be written as: Remainder Theorem: If a polynomial _____ is divided by ______, then the remainder is __________. Synthetic Division: Only use the ___________ of the polynomial and the _____________ must be in the form _________. Factor Theorem: A polynomial _____ has a factor ________ if and only if ___________.

Use polynomial long division EXAMPLE 1 SOLUTION Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x2 in the dividend. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

) Use polynomial long division EXAMPLE 1 3x2 + 4x – 3 x2 – 3x + 5 quotient x2 – 3x + 5 3x4 – 5x3 + 0x2 + 4x – 6 ) Multiply divisor by 3x4/x2 = 3x2 3x4 – 9x3 + 15x2 Subtract. Bring down next term. 4x3 – 15x2 + 4x 4x3 – 12x2 + 20x Multiply divisor by 4x3/x2 = 4x Subtract. Bring down next term. –3x2 – 16x – 6 –3x2 + 9x – 15 Multiply divisor by – 3x2/x2 = – 3 –25x + 9 remainder

Use polynomial long division EXAMPLE 1 3x4 – 5x3 + 4x – 6 x2 – 3x + 5 = 3x2 + 4x – 3 + –25x + 9 ANSWER CHECK You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend. (3x2 + 4x – 3)(x2 – 3x + 5) + (–25x + 9) = 3x2(x2 – 3x + 5) + 4x(x2 – 3x + 5) – 3(x2 – 3x + 5) – 25x + 9 = 3x4 – 9x3 + 15x2 + 4x3 – 12x2 + 20x – 3x2 + 9x – 15 – 25x + 9 = 3x4 – 5x3 + 4x – 6

) Use polynomial long division with a linear divisor EXAMPLE 2 x2 + 7x quotient + 7 x – 2 x3 + 5x2 – 7x + 2 ) x3 – 2x2 Multiply divisor by x3/x = x2. 7x2 – 7x Subtract. 7x2 – 14x Multiply divisor by 7x2/x = 7x. 7x + 2 Subtract. 7x – 14 Multiply divisor by 7x/x = 7. remainder 16 ANSWER x3 + 5x2 – 7x +2 x – 2 = x2 + 7x + 7 + 16

for Examples 1 and 2 GUIDED PRACTICE Divide using polynomial long division. (2x2 – 3x + 8) + –18x + 7 x2 + 2x – 1 ANSWER (x2 – 3x + 10) + –30 x + 2 ANSWER

Use synthetic division EXAMPLE 3 SOLUTION –3 2 1 –8 5 –6 15 –21 2 –5 7 –16 2x3 + x2 – 8x + 5 x + 3 = 2x2 – 5x + 7 – 16 ANSWER

Factor a polynomial EXAMPLE 4 SOLUTION Because x + 2 is a factor of f (x), you know that f (–2) = 0. Use synthetic division to find the other factors. –2 3 –4 –28 –16 –6 20 16 3 –10 –8 0

Use the result to write f (x) as a product of two Factor a polynomial EXAMPLE 4 Use the result to write f (x) as a product of two factors and then factor completely. f (x) = 3x3 – 4x2 – 28x – 16 Write original polynomial. = (x + 2)(3x2 – 10x – 8) Write as a product of two factors. = (x + 2)(3x + 2)(x – 4) Factor trinomial.

for Examples 3 and 4 GUIDED PRACTICE Divide using synthetic division. Factor the polynomial completely given that x – 4 is a factor. x2 + x – 4 + 11 x + 3 ANSWER ANSWER (x – 4)(x –3)(x + 1) 4x2 + 5x + 2 + 9 x – 1 ANSWER ANSWER (x – 4)(x –2)(x +5)

Standardized Test Practice EXAMPLE 5 SOLUTION Because f (3) = 0, x – 3 is a factor of f (x). Use synthetic division. 3 1 –2 –23 60 3 3 –60 1 1 –20 0 Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x3 – 2x2 – 23x + 60 = (x – 3)(x2 + x – 20) = (x – 3)(x + 5)(x – 4) The zeros are 3, –5, and 4. The correct answer is A. ANSWER

for Example 5 GUIDED PRACTICE Find the other zeros of f given that f (–2) = 0. ANSWER 3 and –3 ANSWER 1 and –7

HOmework: Pg. 356 #15-35 eoo