How do find the quotient of #(y^3 - 125)/(y - 5)#?

Question

Abigail Allen · Accepted Answer

#(y^2+5y+25)#

In order to eliminate the denominator, factoring is required. To do this, the difference of cubes must be used.

#a3 + b3 = (a + b)(a2 – ab + b2) #

#a3 – b3 = (a – b)(a2 + ab + b2)#

We intend to employ the latter.

Also #a=y, b=5#

Turn #y^3 -125# to #y^3 - 5^3#

Apply the formula.

#y^3 - 5^3=(y-5)(y^2+5y+25)#

Incorporate it into the numerator.

#((y-5)(y^2+5y+25))/((y-5))#

Simplify

#(y^2+5y+25)#

Hazel Anglin · Answer

undefined To find the quotient of (y^3 - 125)/(y - 5), we can use polynomial long division or synthetic division. Let's use polynomial long division:

Step 1: Divide the first term of the numerator (y^3) by the first term of the denominator (y). The result is y^2.

Step 2: Multiply the entire denominator (y - 5) by the result from step 1 (y^2), and write the product below the numerator.

y^2 * (y - 5) = y^3 - 5y^2

Step 3: Subtract the product obtained in step 2 from the numerator (y^3 - 125) and write the result below the line.

(y^3 - 125) - (y^3 - 5y^2) = -5y^2 - 125

Step 4: Repeat steps 1-3 with the new expression (-5y^2 - 125).

Step 5: Divide the first term of the new expression (-5y^2) by the first term of the denominator (y). The result is -5y.

Step 6: Multiply the entire denominator (y - 5) by the result from step 5 (-5y), and write the product below the line.

-5y * (y - 5) = -5y^2 + 25y

Step 7: Subtract the product obtained in step 6 from the new expression (-5y^2 - 125) and write the result below the line.

(-5y^2 - 125) - (-5y^2 + 25y) = -150y - 125

Step 8: Repeat steps 1-3 with the new expression (-150y - 125).

Step 9: Divide the first term of the new expression (-150y) by the first term of the denominator (y). The result is -150.

Step 10: Multiply the entire denominator (y - 5) by the result from step 9 (-150), and write the product below the line.

-150 * (y - 5) = -150y + 750

Step 11: Subtract the product obtained in step 10 from the new expression (-150y - 125) and write the result below the line.

(-150y - 125) - (-150y + 750) = -875

The quotient of (y^3 - 125)/(y - 5) is y^2 - 5y - 150 with a remainder of -875.