Select all of the factors of x³ + 5x² + 2x - 8;
x + 5, x - 3, x + 4, x - 1, x + 3, x + 2

Solution:

Given the polynomial x³ + 5x² + 2x - 8.

To check if the given options are the linear factors of the given polynomial, let us apply the factor theorem.

If a polynomial f(x) is divisible by (x-a), then the remainder p(a) =0

Step 1: Let us check if (x - 1) as a factor of the polynomial.

⇒ p(1) = (1)³ + 5(1)² + 2(1) - 8

⇒ p(1) = 1+ 5 + 2 - 8

⇒ p(1) = 0

Step 2: Divide the polynomial p(x) = x³ + 5x² + 2x - 8 by the factor g(x) = (x - 1).

⇒ (x³ + 5x² + 2x - 8) ÷ (x - 1)

⇒ x² + 6x + 8

Let us factorize the above polynomial to find the value of x by splitting the middle term.

Step 3: Identify the values of a, b and c.

In the above equation, a is coefficient of x² = 1, b is the coefficient of x = 6 and c is the constant term = - 8.

Step 4: Multiply a and c and find the factors that add up to b.

1 × (8) = 8

⇒ 4 and 2 are the factors that add up to b.

Step 5: Split bx into two terms.

x² + 4x + 2x + 8

Step 6: Take out the common factors by grouping.

x² + 4x + 2x + 8 = x(x + 4) + 2 (x + 4)

=(x + 2) (x + 4)

Thus the factors of the given polynomial = (x + 4) ( x + 2) (x - 1)

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Select all of the factors of x³ + 5x² + 2x - 8;
x + 5, x - 3, x + 4, x - 1, x + 3, x + 2

Summary:

(x + 4), ( x + 2), (x - 1) are the factors of the polynomial x³ + 5x² + 2x - 8 .