Use the remainder theorem to find [tex]$P(k)$[/tex].
[tex]$k=-3 ; P(x)=4 x^3-5 x^2-3 x+11$[/tex]

To find P ( − 3 ) , substitute x = − 3 into the polynomial P ( x ) = 4 x 3 − 5 x 2 − 3 x + 11 . After calculating the values, we find that P ( − 3 ) = − 133 . This uses the remainder theorem, which states that evaluating the polynomial at a point gives the remainder of the polynomial when divided by ( x − k ) .
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Answered by Anonymous | 2025-07-04

Substitute x = − 3 into the polynomial P ( x ) = 4 x 3 − 5 x 2 − 3 x + 11 .
Calculate ( − 3 ) 3 = − 27 and ( − 3 ) 2 = 9 .
Simplify the expression: P ( − 3 ) = 4 ( − 27 ) − 5 ( 9 ) − 3 ( − 3 ) + 11 = − 108 − 45 + 9 + 11 .
Evaluate to find the final answer: − 133 ​ .

Explanation

Understanding the Problem We are given the polynomial P ( x ) = 4 x 3 − 5 x 2 − 3 x + 11 and we want to find P ( k ) where k = − 3 using the remainder theorem. The remainder theorem states that if we divide a polynomial P ( x ) by ( x − k ) , the remainder is P ( k ) . Therefore, we need to evaluate P ( − 3 ) .

Substituting the Value To find P ( − 3 ) , we substitute x = − 3 into the polynomial P ( x ) = 4 x 3 − 5 x 2 − 3 x + 11 . This gives us: P ( − 3 ) = 4 ( − 3 ) 3 − 5 ( − 3 ) 2 − 3 ( − 3 ) + 11

Simplifying the Expression Now, we simplify the expression:


First, we calculate ( − 3 ) 3 = − 27 and ( − 3 ) 2 = 9 .
So, we have: P ( − 3 ) = 4 ( − 27 ) − 5 ( 9 ) − 3 ( − 3 ) + 11 P ( − 3 ) = − 108 − 45 + 9 + 11 P ( − 3 ) = − 153 + 20 P ( − 3 ) = − 133

Final Answer Therefore, P ( − 3 ) = − 133 .

Examples
Polynomials can be used to model various real-world phenomena, such as the trajectory of a projectile or the growth of a population. Evaluating a polynomial at a specific point, as we did here, can help us understand the behavior of the model at that particular value. For instance, if P ( x ) represents the height of a ball at time x , then P ( − 3 ) would give us the height of the ball at time x = − 3 .

Answered by GinnyAnswer | 2025-07-04