Decompose the polynomial into linear factors (x2−7x+6)(x2+3x−18)
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To decompose the given polynomial into linear factors, we need to factorize each quadratic expression in the parentheses.
For the first quadratic expression, x^2 - 7x + 6, we can factorize it as (x - 6)(x - 1).
And for the second quadratic expression, x^2 + 3x - 18, we can factorize it as (x - 3)(x + 6).
Therefore, the polynomial can be decomposed into linear factors as (x - 6)(x - 1)(x - 3)(x + 6).
To decompose the given polynomial into linear factors, we need to factorize each quadratic factor.
Let's start by factorizing the first quadratic factor: x^2-7x+6.
To factorize the quadratic, we need to find two numbers whose product is equal to the product of the square term (x^2) and the constant term (6), and whose sum is equal to the coefficient of the linear term (-7x).
The product of x^2 and 6 is 6x^2, and the sum of the linear term is -7x.
The numbers that satisfy these conditions are -1 and -6 since (-1) * (-6) = 6 and (-1) + (-6) = -7.
Therefore, we can rewrite the first quadratic factor as:
x^2 - 7x + 6 = (x - 1)(x - 6).
Now, let's factorize the second quadratic factor: x^2 + 3x - 18.
Again, we need to find two numbers whose product is equal to the product of the square term (x^2) and the constant term (-18), and whose sum is equal to the coefficient of the linear term (3x).
The product of x^2 and -18 is -18x^2, and the sum of the linear term is 3x.
The numbers that satisfy these conditions are 6 and -3 since 6 * (-3) = -18 and 6 + (-3) = 3.
Therefore, we can rewrite the second quadratic factor as:
x^2 + 3x - 18 = (x + 6)(x - 3).
Now, we can rewrite the given polynomial as a product of these linear factors:
(x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 1)(x - 6)(x + 6)(x - 3).
To decompose the given polynomial into linear factors, (x^2−7x+6)(x^2+3x−18), we first need to factor each quadratic expression.
Step 1: Factor the expression x^2−7x+6.
To factor this quadratic expression, we need to find two numbers that multiply to give us 6 and add up to -7 (the coefficient of the middle term).
We can rewrite the expression as follows:
(x^2−x−6x+6)
To factor, we group the terms:
[(x^2−x)−(6x−6)]
Now, factor out the common factors from each group:
x(x−1)−6(x−1)
Now, notice that we have a common factor (x−1). Factor this out:
(x−1)(x−6)
So, the expression x^2−7x+6 factors as (x−1)(x−6).
Step 2: Factor the expression x^2+3x−18.
To factor this quadratic expression, we need to find two numbers that multiply to give us -18 and add up to 3 (the coefficient of the middle term).
We can rewrite the expression as follows:
(x^2+6x−3x−18)
To factor, we group the terms:
[(x^2+6x)−(3x+18)]
Now, factor out the common factors from each group:
x(x+6)−3(x+6)
Now, notice that we have a common factor (x+6). Factor this out:
(x+6)(x−3)
So, the expression x^2+3x−18 factors as (x+6)(x−3).
Step 3: Combine the factors.
Now, we have the factors of both quadratic expressions:
(x−1)(x−6) and (x+6)(x−3)
To decompose the polynomial into linear factors, we multiply these factors together:
(x−1)(x−6)(x+6)(x−3)
Therefore, the decomposed form of the polynomial (x^2−7x+6)(x^2+3x−18) is (x−1)(x−6)(x+6)(x−3).