Answer :
To determine whether or not a binomial is a factor of a polynomial, we can use the Remainder Theorem. The theorem states that a binomial [tex]\(x - c\)[/tex] is a factor of a polynomial [tex]\(P(x)\)[/tex] if and only if the remainder when [tex]\(P(x)\)[/tex] is divided by [tex]\(x - c\)[/tex] is zero.
Let's evaluate each of the three cases:
1. Is [tex]\(x + 4\)[/tex] a factor of [tex]\(x^4 - 3x^3 - 7x^2 + 5x - 316\)[/tex]?
To check if [tex]\(x + 4\)[/tex] is a factor, evaluate the polynomial at the root of [tex]\(x + 4 = 0\)[/tex], which is [tex]\(x = -4\)[/tex]. If the remainder is 0, then [tex]\(x + 4\)[/tex] is a factor.
From calculations, the remainder is 0:
[tex]\[\text{Remainder} = 0\][/tex]
Conclusion: Yes, [tex]\(x + 4\)[/tex] is a factor because there's no remainder.
2. Is [tex]\(2x-3\)[/tex] a factor of [tex]\(32x^5 - 64x^4 - 72x^3 + 12x^2 - 4x + 303\)[/tex]?
To determine this, set [tex]\(2x - 3 = 0\)[/tex], which gives [tex]\(x = \frac{3}{2}\)[/tex]. Evaluate the polynomial at [tex]\(x = \frac{3}{2}\)[/tex]. If the remainder is 0, then [tex]\(2x - 3\)[/tex] is a factor.
From calculations, the remainder is 0:
[tex]\[\text{Remainder} = 0\][/tex]
Conclusion: Yes, [tex]\(2x - 3\)[/tex] is a factor because there's no remainder.
3. Is [tex]\(x-7\)[/tex] a factor of [tex]\(x^3 + 6x^2 + 3x + 70\)[/tex]?
Set [tex]\(x - 7 = 0\)[/tex], which means [tex]\(x = 7\)[/tex]. Evaluate the polynomial at [tex]\(x = 7\)[/tex]. The polynomial remainder when evaluated should be 0 if [tex]\(x - 7\)[/tex] is a factor.
From calculations, the remainder is:
[tex]\[\text{Remainder} = 728\][/tex]
Conclusion: No, [tex]\(x - 7\)[/tex] is not a factor because the remainder is not zero (it's 728).
These conclusions are based on following the remainder theorem by calculating whether the polynomial provides zero remainder when divided by each binomial.
Let's evaluate each of the three cases:
1. Is [tex]\(x + 4\)[/tex] a factor of [tex]\(x^4 - 3x^3 - 7x^2 + 5x - 316\)[/tex]?
To check if [tex]\(x + 4\)[/tex] is a factor, evaluate the polynomial at the root of [tex]\(x + 4 = 0\)[/tex], which is [tex]\(x = -4\)[/tex]. If the remainder is 0, then [tex]\(x + 4\)[/tex] is a factor.
From calculations, the remainder is 0:
[tex]\[\text{Remainder} = 0\][/tex]
Conclusion: Yes, [tex]\(x + 4\)[/tex] is a factor because there's no remainder.
2. Is [tex]\(2x-3\)[/tex] a factor of [tex]\(32x^5 - 64x^4 - 72x^3 + 12x^2 - 4x + 303\)[/tex]?
To determine this, set [tex]\(2x - 3 = 0\)[/tex], which gives [tex]\(x = \frac{3}{2}\)[/tex]. Evaluate the polynomial at [tex]\(x = \frac{3}{2}\)[/tex]. If the remainder is 0, then [tex]\(2x - 3\)[/tex] is a factor.
From calculations, the remainder is 0:
[tex]\[\text{Remainder} = 0\][/tex]
Conclusion: Yes, [tex]\(2x - 3\)[/tex] is a factor because there's no remainder.
3. Is [tex]\(x-7\)[/tex] a factor of [tex]\(x^3 + 6x^2 + 3x + 70\)[/tex]?
Set [tex]\(x - 7 = 0\)[/tex], which means [tex]\(x = 7\)[/tex]. Evaluate the polynomial at [tex]\(x = 7\)[/tex]. The polynomial remainder when evaluated should be 0 if [tex]\(x - 7\)[/tex] is a factor.
From calculations, the remainder is:
[tex]\[\text{Remainder} = 728\][/tex]
Conclusion: No, [tex]\(x - 7\)[/tex] is not a factor because the remainder is not zero (it's 728).
These conclusions are based on following the remainder theorem by calculating whether the polynomial provides zero remainder when divided by each binomial.
