Question

The polynomial p(x) = x^3 - 21x - 20 has a known factor of (x-5). Rewrite p(x) as a product of linear factors.

Answer 1

Answer:

p(x) = (x - 5)(x + 1)(x + 4)

Step-by-step explanation:

Using Synthetic division to find the quotient on dividing p(x) by (x - 5)

5 | 1  0  - 21   - 20 ← 0 to represent the coefficient of x² term

        5    25     20

    -----------------------

     1  5   4        0 ← remainder is zero

quotient = x² + 5x + 4 = (x + 1)(x + 4)

p(x) = (x - 5)(x + 1)(x + 4)

Answer 2

The polynomial p(x) = x^3 - 21x - 20 can be written as a product of linear factors as follows;

(x-5) (x+4) (x+1).

According to the question, the polynomial, p(x) = x^3 - 21x - 20 has a known factor of (x- 5).

• To further factorise this polynomial such that it becomes a product of linear factors, we must first, perform the long division of x³ - 21x - 20 divided by (x-5).

The polynomial long division is as in the image attached;

The result of the division of x³ - 21x - 20 by (x-5) yields;

• x² + 5x + 4.

The expression, x² + 5x + 4 can however be further factorised so that we have;

• x² + 5x + 4 = (x+4) (x-1)

Ultimately, the polynomial p(x) = x^3 - 21x - 20 can be written as a product of linear factors as follows;

(x-5) (x+4) (x+1).

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