What is the remainder when the polynomial (see picture) is divided by x+5?

60

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To get this answer, we plug x = -5 into the polynomial function below. This is using the remainder theorem which says "dividing p(x) by x-k will lead to a remainder of r = p(k)". In this case, x+5 = x-(-5) = x-k so k = 5.

Replace every x with -5 to get...

p(x) = 5x^2 + 10x - 15

p(-5) = 5(-5)^2 + 10(-5) - 15

p(-5) =5(25) + 10(-5) - 15

p(-5) = 125 - 50 - 15

p(-5) = 75 - 15

p(-5) = 60

Another way to get this answer is to use either polynomial long division (see figure 1) or synthetic division (see figure 2). Both figures are attached as images below.

an

y = 6x -3y -12

y +3y +12 = 6x

=> . x = { 2y / 3 +2 } this is equation of line in terms of x

=> . if line meets the x-axis the co-ordinates of meeting point would be ( x , 0 )

=> . x = 2 ( 0 ) / 3 + 2

=> . x = 2

so, the point is ( 2 , 0 )

answer is 8x+24

area of vanessa's patio = length × width

given, length = (3x² + 5x + 10)

width = (x² - 3x - 1)

area of vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)

= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)

= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10

= 3x⁴ - 4x³ - 8x² - 35x - 10

hence, expression of vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)