Answer :
To solve this problem, we will use the Remainder Theorem, which states that if a polynomial [tex]\( p(t) \)[/tex] is divided by a linear divisor [tex]\( (t-a) \)[/tex], the remainder of that division is [tex]\( p(a) \)[/tex].
Given the polynomial [tex]\( t^3 - 3t^2 + kt + 50 \)[/tex], and the information that when this polynomial is divided by [tex]\( (t-3) \)[/tex], the remainder is 62, we can use the theorem as follows:
1. According to the Remainder Theorem, the remainder when dividing by [tex]\( (t-3) \)[/tex] is equal to the value of the polynomial evaluated at [tex]\( t = 3 \)[/tex]. That is [tex]\( p(3) = 62 \)[/tex].
2. Substitute [tex]\( t = 3 \)[/tex] into the polynomial:
[tex]\[
p(3) = 3^3 - 3(3)^2 + k \cdot 3 + 50
\][/tex]
3. Calculate each term:
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 3(3)^2 = 27 \)[/tex]
4. Substitute these into the expression:
[tex]\[
p(3) = 27 - 27 + 3k + 50
\][/tex]
5. Simplify the equation:
[tex]\[
p(3) = 0 + 3k + 50
\][/tex]
[tex]\[
p(3) = 3k + 50
\][/tex]
6. Set this expression equal to 62, since [tex]\( p(3) = 62 \)[/tex]:
[tex]\[
3k + 50 = 62
\][/tex]
7. Solve for [tex]\( k \)[/tex]:
- Subtract 50 from both sides:
[tex]\[
3k = 62 - 50
\][/tex]
[tex]\[
3k = 12
\][/tex]
- Divide by 3:
[tex]\[
k = \frac{12}{3}
\][/tex]
[tex]\[
k = 4
\][/tex]
So the value of [tex]\( k \)[/tex] is 4.
Given the polynomial [tex]\( t^3 - 3t^2 + kt + 50 \)[/tex], and the information that when this polynomial is divided by [tex]\( (t-3) \)[/tex], the remainder is 62, we can use the theorem as follows:
1. According to the Remainder Theorem, the remainder when dividing by [tex]\( (t-3) \)[/tex] is equal to the value of the polynomial evaluated at [tex]\( t = 3 \)[/tex]. That is [tex]\( p(3) = 62 \)[/tex].
2. Substitute [tex]\( t = 3 \)[/tex] into the polynomial:
[tex]\[
p(3) = 3^3 - 3(3)^2 + k \cdot 3 + 50
\][/tex]
3. Calculate each term:
- [tex]\( 3^3 = 27 \)[/tex]
- [tex]\( 3(3)^2 = 27 \)[/tex]
4. Substitute these into the expression:
[tex]\[
p(3) = 27 - 27 + 3k + 50
\][/tex]
5. Simplify the equation:
[tex]\[
p(3) = 0 + 3k + 50
\][/tex]
[tex]\[
p(3) = 3k + 50
\][/tex]
6. Set this expression equal to 62, since [tex]\( p(3) = 62 \)[/tex]:
[tex]\[
3k + 50 = 62
\][/tex]
7. Solve for [tex]\( k \)[/tex]:
- Subtract 50 from both sides:
[tex]\[
3k = 62 - 50
\][/tex]
[tex]\[
3k = 12
\][/tex]
- Divide by 3:
[tex]\[
k = \frac{12}{3}
\][/tex]
[tex]\[
k = 4
\][/tex]
So the value of [tex]\( k \)[/tex] is 4.
