2x^3 − 7x^2 − 17x + 10 = 0; 1/ 2

Question

A partial solution set is given for the equation. Find the complete solution set. (Enter your answers as a comma-separated list.)

Answer 1

The function is , and the root is 1/2.

Use synthetic division to find .

Step 1: Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients as shown at the right.

Step 2: Write the constant r of the divisor (x - r) to the left. In this case, . Bring the first coefficient, 2, down.

Step 3: Multiply the first coefficient by r : . Write the product under the second coefficient, - 7 and : .

Step 4: Multiply the sum, , by r : .

Write the product under the next coefficient, - 17 and add : .

Step 5: Multiply the sum, , by r : .

Write the product under the next coefficient, 10 and add : . The remainder is 0.

The numbers along the bottom row are the coefficients of the quotient. Start with the power of x that is one less than the degree of the dividend. Thus, the quotient is .

The resulting equation is 2x² - 6x - 20 = 0 and solve for x by using factors method.

To find the coefficients of the x - terms, you must find two numbers with a product of 2 · (- 20) or - 40, and a sum of - 6. The two coefficients must be - 10 and 4 since  (- 10)(4)  = - 40 and - 10 + 4  = - 6.

Rewrite the equation using - 10x and 4x in place of - 6x and factor by grouping.

Substitute - 10x + 4x for -6x in 2x² - 6x - 20 = 0.

2x² - 10x + 4x - 20 = 0

(2x² - 10x) + (4x - 20) = 0

Factor out the GCF of each group.

2x(x - 5) + 4(x - 5) = 0

(x - 5) is the common factor.

(2x + 4)(x - 5) = 0

2(x + 2)(x - 5) = 0.

(x + 2)(x - 5) = 0.

Apply Zero Product Property :  For any real numbers a and b, if ab = 0, then either a = 0, b = 0, or both a and b equal zero.

x + 2 = 0 and x - 5 = 0

x = - 2 and x = 5.

The solution set is {- 2, 1/2, 5}.