Answer :
Let's go through the problem step-by-step to identify the errors and to determine whether [tex]\( x-5 \)[/tex] is indeed a factor of the polynomial [tex]\(-3x^4 + 15x^3 - x + 5\)[/tex].
Step 1: Setting up the polynomial division
The polynomial we are considering is:
[tex]\[ -3x^4 + 15x^3 - x + 5 \][/tex]
We are dividing this polynomial by:
[tex]\[ x - 5 \][/tex]
Step 2: Performing polynomial division
When we divide a polynomial by a binomial of the form [tex]\( x - k \)[/tex], we use either long division or synthetic division. For clarity, let's explain the general process of polynomial long division:
1. Divide the first term of the dividend by the first term of the divisor.
2. Multiply the entire divisor by this result and subtract from the dividend.
3. Repeat the process with the new polynomial formed.
Result and verification
Given the results of the division:
[tex]\[ -3x^4 + 15x^3 - x + 5 \quad \div \quad x - 5 \][/tex]
The quotient (result of the division) is:
[tex]\[ -3x^3 - 1 \][/tex]
And the remainder is:
[tex]\[ 0 \][/tex]
Step 3: Analyzing the errors
The student concluded that [tex]\( x-5 \)[/tex] is not a factor of the polynomial.
Error 1: Misunderstanding the Remainder
The first error the student made is in concluding that [tex]\( x-5 \)[/tex] is not a factor based on the remainder. According to the Remainder Theorem, for [tex]\( x-k \)[/tex] to be a factor of a polynomial [tex]\( P(x) \)[/tex], the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x-k \)[/tex] must be zero.
In this case, the remainder is 0:
[tex]\[ P(x) = (-3x^4 + 15x^3 - x + 5) \quad \div \quad (x - 5) = -3x^3 - 1 \quad \text{with remainder} \quad 0 \][/tex]
Since the remainder is 0, [tex]\( x-5 \)[/tex] is indeed a factor of [tex]\(-3x^4 + 15x^3 - x + 5\)[/tex].
Error 2: Incorrect Conclusion Based on the Quotient
The second error relates to possibly misunderstanding how to interpret the quotient in polynomial division. The correctness of the quotient [tex]\(-3x^3 - 1\)[/tex] does not impact the determination of whether [tex]\( x-5 \)[/tex] is a factor; this determination solely depends on the remainder being 0.
Step 4: Conclusion
Given that the remainder is indeed 0 from the division, we conclude:
[tex]\[ x - 5 \text{ is a factor of the polynomial } -3x^4 + 15x^3 - x + 5. \][/tex]
In summary:
1. The student made an error in not recognizing that a remainder of 0 means [tex]\( x-5 \)[/tex] is a factor.
2. The quotient [tex]\(-3x^3 - 1\)[/tex] and the remainder 0 confirm that [tex]\( x-5 \)[/tex] is indeed a factor of the given polynomial.
Step 1: Setting up the polynomial division
The polynomial we are considering is:
[tex]\[ -3x^4 + 15x^3 - x + 5 \][/tex]
We are dividing this polynomial by:
[tex]\[ x - 5 \][/tex]
Step 2: Performing polynomial division
When we divide a polynomial by a binomial of the form [tex]\( x - k \)[/tex], we use either long division or synthetic division. For clarity, let's explain the general process of polynomial long division:
1. Divide the first term of the dividend by the first term of the divisor.
2. Multiply the entire divisor by this result and subtract from the dividend.
3. Repeat the process with the new polynomial formed.
Result and verification
Given the results of the division:
[tex]\[ -3x^4 + 15x^3 - x + 5 \quad \div \quad x - 5 \][/tex]
The quotient (result of the division) is:
[tex]\[ -3x^3 - 1 \][/tex]
And the remainder is:
[tex]\[ 0 \][/tex]
Step 3: Analyzing the errors
The student concluded that [tex]\( x-5 \)[/tex] is not a factor of the polynomial.
Error 1: Misunderstanding the Remainder
The first error the student made is in concluding that [tex]\( x-5 \)[/tex] is not a factor based on the remainder. According to the Remainder Theorem, for [tex]\( x-k \)[/tex] to be a factor of a polynomial [tex]\( P(x) \)[/tex], the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( x-k \)[/tex] must be zero.
In this case, the remainder is 0:
[tex]\[ P(x) = (-3x^4 + 15x^3 - x + 5) \quad \div \quad (x - 5) = -3x^3 - 1 \quad \text{with remainder} \quad 0 \][/tex]
Since the remainder is 0, [tex]\( x-5 \)[/tex] is indeed a factor of [tex]\(-3x^4 + 15x^3 - x + 5\)[/tex].
Error 2: Incorrect Conclusion Based on the Quotient
The second error relates to possibly misunderstanding how to interpret the quotient in polynomial division. The correctness of the quotient [tex]\(-3x^3 - 1\)[/tex] does not impact the determination of whether [tex]\( x-5 \)[/tex] is a factor; this determination solely depends on the remainder being 0.
Step 4: Conclusion
Given that the remainder is indeed 0 from the division, we conclude:
[tex]\[ x - 5 \text{ is a factor of the polynomial } -3x^4 + 15x^3 - x + 5. \][/tex]
In summary:
1. The student made an error in not recognizing that a remainder of 0 means [tex]\( x-5 \)[/tex] is a factor.
2. The quotient [tex]\(-3x^3 - 1\)[/tex] and the remainder 0 confirm that [tex]\( x-5 \)[/tex] is indeed a factor of the given polynomial.
