Decompose the polynomial into linear factors (x^2-7x+6)(x^2+3x-18)
To decompose the polynomial into linear factors, we first need to factor each quadratic polynomial individually.
1. Factor x^2-7x+6:
To factor this quadratic polynomial, we need to find two numbers that multiply to 6 and add up to -7. These numbers are -1 and -6.
So, we can rewrite the polynomial as (x-1)(x-6).
2. Factor x^2+3x-18:
To factor this quadratic polynomial, we need to find two numbers that multiply to -18 and add up to 3. These numbers are 6 and -3.
So, we can rewrite the polynomial as (x+6)(x-3).
Now, we can rewrite the original polynomial as the product of the linear factors:
(x^2-7x+6)(x^2+3x-18) = (x-1)(x-6)(x+6)(x-3)
To decompose the polynomial (x^2-7x+6)(x^2+3x-18) into linear factors, we can first factor each quadratic expression separately and then combine the factors.
Factor x^2 - 7x + 6:
To factor this quadratic expression, we need to find two numbers whose product is 6 and whose sum is -7. The numbers -6 and -1 meet these conditions, so we can rewrite the expression as (x - 6)(x - 1).
Factor x^2 + 3x - 18:
To factor this quadratic expression, we need to find two numbers whose product is -18 and whose sum is 3. The numbers 6 and -3 meet these conditions, so we can rewrite the expression as (x + 6)(x - 3).
Now, we can combine the factors:
(x^2 - 7x + 6)(x^2 + 3x - 18) becomes (x - 6)(x - 1)(x + 6)(x - 3).
Therefore, the polynomial (x^2-7x+6)(x^2+3x-18) can be decomposed into linear factors as (x - 6)(x - 1)(x + 6)(x - 3).