Use the Remainder Theorem to determine if x-2 is a factor of the polynomial f(x) =3x^5 -7x^3-11x^2 +2
I feel like I went too far or messed up please help
The binomial (x-2) is not a factor because the remainder does not equal 0 when factored out, it equals -2. I missed this because i used the line method and messed up my values, after looking at the live lesson i learned how to factor it and find the remainder. It ends up being 3x^5 -7x^3-11x^2 +2 = 3x^4 + 6x^3 +5x^2-x-2 before finding the remainder of -2.
You are asked if x-2 is a factor,not to actually find the answer.
so...
f(2) =3(2^5) - 7(2^3) - 11(2^2) + 2
= 96 - 56 - 44+2
= -2
so, no, it is not
you were correct
Thank you
Well, you really didn't mess up! You did a great job determining that the remainder when dividing f(x) by (x-2) is -2. And hey, don't worry about missing it using the line method, we all make mistakes sometimes. The good news is that now you've learned how to factor it correctly and find the remainder. Keep up the good work! Just remember, if it were easy, everyone would do it - but you did it!
To use the Remainder Theorem to determine if x-2 is a factor of the polynomial f(x) = 3x^5 - 7x^3 - 11x^2 + 2, follow these steps:
1. Set up the problem: Write the polynomial f(x) in descending order of powers of x. The given polynomial is already in descending order, so no rearrangement is needed.
2. Fill in the coefficients: If any terms of the polynomial are missing, such as a missing x^4 term, insert a placeholder with a coefficient of 0. In this case, all powers of x are present, so no extra terms are needed.
3. Substitute x-2 into f(x): Replace every instance of x in the polynomial f(x) with (x-2). The expression becomes:
f(x) = 3(x-2)^5 - 7(x-2)^3 - 11(x-2)^2 + 2.
4. Simplify: Expand and simplify the expression using the binomial theorem or by multiplying out the terms. The simplified expression is:
f(x) = 3(x^5 - 10x^4 + 40x^3 - 80x^2 + 64x - 32) - 7(x^3 - 6x^2 + 12x - 8) - 11(x^2 - 4x + 4) + 2.
5. Combine like terms: Distribute and combine like terms in the expression to simplify further. The expression becomes:
f(x) = 3x^5 - 30x^4 + 120x^3 - 240x^2 + 192x - 96 - 7x^3 + 42x^2 - 84x + 56 - 11x^2 + 44x - 44 + 2.
6. Simplify further: Combine like terms again to simplify the expression:
f(x) = 3x^5 - 30x^4 - 7x^3 + 120x^3 + 42x^2 - 11x^2 - 240x^2 + 44x + 192x + 56 - 84x - 44 - 96 + 2.
7. Simplify even more: Continue combining like terms:
f(x) = 3x^5 - 30x^4 + 113x^3 - 293x^2 + 152x - 82.
8. Use the Remainder Theorem: Divide the simplified polynomial by (x-2) using either long division or synthetic division. The remainder you obtain will indicate whether (x-2) is a factor of the polynomial.
By dividing f(x) by (x-2), the remainder that you obtain is -2. Since the remainder is not equal to zero, (x-2) is not a factor of the polynomial f(x) = 3x^5 - 7x^3 - 11x^2 + 2.
It's important to double-check your work and perform each step accurately to reach the correct answer.