<p>Remainder and Factor Theorem</p><p>Example 1 SKIING The formula for distance traveled by a skier t seconds into a ski run is 2 d(t) = v0t + at , where d(t) is the distance traveled, v0 is the initial velocity, and a is the acceleration. Find the distance traveled after 60 seconds if the skier’s initial velocity is 4 meters per second with an acceleration rate of 1.0 meters per second squared.</p><p>The distance formula becomes d(t) = 4t + (1.0)t2 or 0.5t2 + 4t. You can use one of two methods to find the distance after 60 seconds.</p><p>Method 1 Method 2 Divide 0.5t2 + 4t by t - 60. Evaluate d(t) for t = 60.</p><p>0.5 t + 34 d(t) = 0.5t2 + 4t t - 60 )0.5t2 + 4t d(60) = 0.5(60)2 + 4(60) 0.5 t 2 -30 t = 0.5(3600) + 4(60) 34t + 0 = 2040 34 t - 2040 2040  d(60) = 2040</p><p>The skier will travel 2040 meters in 60 seconds.</p><p>Example 2 Divide 4x3 - 9x2 - 8x - 3 by x - 3 using synthetic division.</p><p>3 4 -9 -8 -3 12 9 3 4 3 1 0</p><p>The quotient is 4x2 + 3x + 1 with a remainder of 0.</p><p>Remainder and Factor Theorem</p><p>Example 3 Use the Remainder Theorem to find the remainder when 4x3 - 3x2 + 7x + 10 is divided by x - 3. State whether the binomial is a factor of the polynomial. Explain.</p><p>Find f(3) to see if x - 3 is a factor. f(x) = 4x3 - 3x2 + 7x + 10 f(3) = 4(33) - 3(32) + 7(3) + 10 Replace x with 3. = 4(27) - 3(9) + 21 + 10 or 112</p><p>Since f(3) = 112, the remainder is not zero. So the binomial (x - 3) is not a factor of the polynomial.</p><p>Example 4 Determine the binomial factors of x3 - 3x2 - 6x + 8.</p><p>Method 1 Use synthetic Division</p><p> r 1 -3 -6 8 -3 1 -6 12 -28 -2 1 -5 4 0 x+ 2 is a factor. -1 1 -4 -2 10 0 1 -3 -6 8 1 1 -2 -8 0 x - 1 is a factor 2 1 -1 -8 -8 3 1 0 -6 -10 4 1 1 -2 0 x - 4 is a factor.</p><p>Method 2 Test some values using the Factor Theorem. f(x) = x3 - 3x2 - 6x + 8 f(2) = 23 - 3(22) - 6(2) + 8 or -8 f(1) = 13 - 3(12) - 6(1) + 8 or 0</p><p>Because f(1) = 0, x - 1 is a factor. Find the depressed polynomial.</p><p>1 1 -3 -6 8 1 -2 -8 1 -2 -8 0</p><p>The depressed polynomial is x2 - 2x - 8. Factor the depressed polynomial. x2 - 2x - 8 = (x - 4)(x + 2)</p><p>The factors of the polynomial are x - 1, x - 4, and x + 2.</p><p>Remainder and Factor Theorem</p><p>Example 5 Find the value of k so that the remainder of (3x4 + 8x3 - 2x2 - kx + 4)  (x + 2) is 0.</p><p>If the remainder is to be 0, x + 2 must be a factor of 3x4 + 8x3 - 2x2 - kx + 4. So, f(-2) must equal 0.</p><p> f(x) = 3x4 + 8x3 - 2x2 - kx + 4 f(-2) = 3(-2)4 + 8(-2)3 - 2(-2)2 - k(-2) + 4 0 = 48 - 64 - 8 + 2k + 4 Replace f(-2) with 0. 0 = 2k - 20 10 = k</p><p>The value of k is 10. Check using synthetic division. -2 3 8 -2 -10 4 -6 -4 12 -4 3 2 -6 2 0 </p>