Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To determine whether [tex]\( x - 1 \)[/tex] is a factor of [tex]\( P(x) = -x^4 + x^3 + 6x^2 - 9 \)[/tex], we can use the Factor Theorem. The Factor Theorem states that [tex]\( x - c \)[/tex] is a factor of the polynomial [tex]\( P(x) \)[/tex] if and only if [tex]\( P(c) = 0 \)[/tex].
In this case, we need to evaluate [tex]\( P(1) \)[/tex] and check if it equals zero.
First, we evaluate the polynomial at [tex]\( x = 1 \)[/tex]:
[tex]\[ P(1) = - (1)^4 + (1)^3 + 6(1)^2 - 9 \][/tex]
Calculate each term step-by-step:
[tex]\[ (1)^4 = 1 \implies - (1)^4 = -1 \][/tex]
[tex]\[ (1)^3 = 1 \][/tex]
[tex]\[ 6(1)^2 = 6 \][/tex]
[tex]\[ -9 \text{ (constant term)} \][/tex]
Combine all the terms:
[tex]\[ P(1) = -1 + 1 + 6 - 9 \][/tex]
Simplify the expression:
[tex]\[ P(1) = 7 - 9 = -3 \][/tex]
So, [tex]\( P(1) = -3 \)[/tex].
Since [tex]\( P(1) \neq 0 \)[/tex], by the Factor Theorem, [tex]\( x - 1 \)[/tex] is not a factor of [tex]\( P(x) \)[/tex].
Thus, the detailed answer is:
[tex]\[ P(1) = -3 \][/tex]
And
[tex]\[ x-1 \text{ is not a factor of } P(x). \][/tex]
In this case, we need to evaluate [tex]\( P(1) \)[/tex] and check if it equals zero.
First, we evaluate the polynomial at [tex]\( x = 1 \)[/tex]:
[tex]\[ P(1) = - (1)^4 + (1)^3 + 6(1)^2 - 9 \][/tex]
Calculate each term step-by-step:
[tex]\[ (1)^4 = 1 \implies - (1)^4 = -1 \][/tex]
[tex]\[ (1)^3 = 1 \][/tex]
[tex]\[ 6(1)^2 = 6 \][/tex]
[tex]\[ -9 \text{ (constant term)} \][/tex]
Combine all the terms:
[tex]\[ P(1) = -1 + 1 + 6 - 9 \][/tex]
Simplify the expression:
[tex]\[ P(1) = 7 - 9 = -3 \][/tex]
So, [tex]\( P(1) = -3 \)[/tex].
Since [tex]\( P(1) \neq 0 \)[/tex], by the Factor Theorem, [tex]\( x - 1 \)[/tex] is not a factor of [tex]\( P(x) \)[/tex].
Thus, the detailed answer is:
[tex]\[ P(1) = -3 \][/tex]
And
[tex]\[ x-1 \text{ is not a factor of } P(x). \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.