How do you use the remainder theorem to determine the remainder when $ 3{t^2} + 5t - 7 $ is divided by $ t - 5 $ ?

Hint: In order to determine the remainder using the remainder theorem for the above division, identify the value for $ a $ by comparing $ t - a $ with the linear factor given. Now Put this value of $ a $ in all the occurrences of $ t $ in the quadratic polynomial to obtain the remainder of the division.

Complete step by step solution:
We are given a quadratic polynomial in variable $ \left( t \right) $ as $ 3{t^2} + 5t - 7 $ which is divided by $ t - 5 $ .
In this question, we have to find out the remainder of the division without actually dividing the polynomial by long division but by applying the remainder theorem.
Before Jumping on to the solution, let’s understand what the remainder theorem actually is and what it states.
So, According to the Remainder Theorem, When a polynomial , let it be called $ p\left( x \right) $ is divided by some linear factor of the form $ \left( {x - a} \right)\, $ , then the remainder is actually a value of $ p\left( x \right) $ at $ x = a $ , specially $ p\left( a \right) $ .
As per the question ,we have a polynomial $ p\left( t \right) = 3{t^2} + 5t - 7 $ which is divided by the linear factor $ t - 5 $ .Here $ a = 5 $ . So, by remainder theorem, we have the remainder of this division as $ p\left( 5 \right) $ which can be calculated by substituting all the occurrences of $ t $ with $ 5 $ , we get
 $
  p\left( 5 \right) = 3{\left( 5 \right)^2} + 5\left( 5 \right) - 7 \\
   = 3\left( {25} \right) + 25 - 7 \\
   = 75 + 18 \\
   = 93 \;
  $
Therefore, the remainder is equal to $ 93 $ when polynomial $ 3{t^2} + 5t - 7 $ is divided by $ t - 5 $ .
So, the correct answer is “93”.

Note: 1. You must know that the remainder theorem is only valid or applied when a function is divided by a linear polynomial. Linear polynomials should be in the form (x + number) or (x – number). For example: $ \left( {x - a} \right)\,or\,\left( {x + a} \right) $ where $ a $ can be any integer value.
2. You can verify your answer by dividing the polynomial using long division method