Question
Decompose the polynomial into linear factors
(x^2-7x+6) (x^2+3x-18)
.
A
(x−6)(x−1)(x+6)(x+3)\left(x-6\right)\left(x-1\right)\left(x+6\right)\left(x+3\right)(x−6)(x−1)(x+6)(x+3)
B
(x+6)(x+1)(x−6)(x+3)\left(x+6\right)\left(x+1\right)\left(x-6\right)\left(x+3\right)(x+6)(x+1)(x−6)(x+3)
C
(x−6)(x−1)(x+6)(x−3)\left(x-6\right)\left(x-1\right)\left(x+6\right)\left(x-3\right)(x−6)(x−1)(x+6)(x−3)
D
(x−6)(x+1)(x−3)(x−6)\left(x-6\right)\left(x+1\right)\left(x-3\right)\left(x-6\right)(x−6)(x+1)(x−3)(x−6)
Answers
GPT-4o mini
To decompose the polynomial \( (x^2 - 7x + 6)(x^2 + 3x - 18) \) into linear factors, we need to factor each quadratic expression.
### Step 1: Factor \( x^2 - 7x + 6 \)
To factor \( x^2 - 7x + 6 \), we look for two numbers that multiply to 6 (the constant term) and add up to -7 (the coefficient of the linear term). The numbers -6 and -1 satisfy these conditions:
\[
x^2 - 7x + 6 = (x - 6)(x - 1)
\]
### Step 2: Factor \( x^2 + 3x - 18 \)
Now, we factor \( x^2 + 3x - 18 \). We look for two numbers that multiply to -18 (the constant term) and add up to 3 (the coefficient of the linear term). The numbers 6 and -3 satisfy these conditions:
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]
### Step 3: Combine the factors
Now we can combine the factors from both quadratics:
\[
(x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 6)(x - 1)(x + 6)(x - 3)
\]
### Final Result
Thus, the complete factorization of the polynomial \( (x^2 - 7x + 6)(x^2 + 3x - 18) \) into linear factors is:
\[
(x - 6)(x - 1)(x + 6)(x - 3)
\]
### Answer
Looking at the options, none seem exactly correct at first glance because \( (x - 3) \) is not included in the provided options, but the format should resemble the factors I derived.
However, if I need to select the closest option that includes similar factors, I will suggest that **option A** \( (x-6)(x-1)(x+6)(x+3) \) is still incorrect because it has \( (x+3) \) where it should be \( (x-3) \).
Thus, the correct factorization is as derived above.
### Step 1: Factor \( x^2 - 7x + 6 \)
To factor \( x^2 - 7x + 6 \), we look for two numbers that multiply to 6 (the constant term) and add up to -7 (the coefficient of the linear term). The numbers -6 and -1 satisfy these conditions:
\[
x^2 - 7x + 6 = (x - 6)(x - 1)
\]
### Step 2: Factor \( x^2 + 3x - 18 \)
Now, we factor \( x^2 + 3x - 18 \). We look for two numbers that multiply to -18 (the constant term) and add up to 3 (the coefficient of the linear term). The numbers 6 and -3 satisfy these conditions:
\[
x^2 + 3x - 18 = (x + 6)(x - 3)
\]
### Step 3: Combine the factors
Now we can combine the factors from both quadratics:
\[
(x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 6)(x - 1)(x + 6)(x - 3)
\]
### Final Result
Thus, the complete factorization of the polynomial \( (x^2 - 7x + 6)(x^2 + 3x - 18) \) into linear factors is:
\[
(x - 6)(x - 1)(x + 6)(x - 3)
\]
### Answer
Looking at the options, none seem exactly correct at first glance because \( (x - 3) \) is not included in the provided options, but the format should resemble the factors I derived.
However, if I need to select the closest option that includes similar factors, I will suggest that **option A** \( (x-6)(x-1)(x+6)(x+3) \) is still incorrect because it has \( (x+3) \) where it should be \( (x-3) \).
Thus, the correct factorization is as derived above.