Find the remainder when P(x) = 3x4 - 2x3 + 5x − 7 is divided by x+1. Let P(x) = x3 + ax2 + bx + c. Find a,b,c…

Question

Find the remainder when P(x) = 3x4 - 2x3 + 5x − 7 is divided by x+1. Let P(x) = x3 + ax2 + bx + c. Find a,b,c if P(1)=6, P(2)=15, P(3)=28.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for problems 9 and 10.

: Find the remainder when $P(x) = 3x^4 - 2x^3 + 5x - 7$ is divided by $x+1$ .

Apply the Remainder Theorem.
According to the Remainder Theorem, if a polynomial $P(x)$ is divided by $(x-a)$ , the remainder is $P(a)$ .
Here, the divisor is $x+1$ , so $a=-1$ . The remainder is $P(-1)$ .
Substitute $x=-1$ into $P(x)$ .
$P(-1) = 3(-1)^4 - 2(-1)^3 + 5(-1) - 7$ $P(-1) = 3(1) - 2(-1) + 5(-1) - 7$ $P(-1) = 3 + 2 - 5 - 7$ $P(-1) = 5 - 5 - 7$ $P(-1) = 0 - 7$ $P(-1) = -7$
The remainder is $\boxed{-7}$ .

: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. Suppose that $P(1)=6$ , $P(2)=15$ , $P(3)=28$ . Find the values of $a$ , $b$ , and $c$ . Hence determine the polynomial $P(x)$ .
Set up a system of equations using the given conditions.
Given $P(x) = x^3 + ax^2 + bx + c$ :
For $P(1)=6$ :
$1^3 + a(1)^2 + b(1) + c = 6$ $1 + a + b + c = 6$ $a + b + c = 5 \quad (1)$
For $P(2)=15$ :
$2^3 + a(2)^2 + b(2) + c = 15$ $8 + 4a + 2b + c = 15$ $4a + 2b + c = 7 \quad (2)$
For $P(3)=28$ :
$3^3 + a(3)^2 + b(3) + c = 28$ $27 + 9a + 3b + c = 28$ $9a + 3b + c = 1 \quad (3)$
Solve the system of equations.
Subtract (1) from (2):
$(4a + 2b + c) - (a + b + c) = 7 - 5$ $3a + b = 2 \quad (4)$
Subtract (2) from (3):
$(9a + 3b + c) - (4a + 2b + c) = 1 - 7$ $5a + b = -6 \quad (5)$
Subtract (4) from (5):
$(5a + b) - (3a + b) = -6 - 2$ $2a = -8$ $a = -4$
Substitute $a=-4$ into equation (4):
$3(-4) + b = 2$ $-12 + b = 2$ $b = 14$
Substitute $a=-4$ and $b=14$ into equation (1):
$-4 + 14 + c = 5$ $10 + c = 5$ $c = -5$
State the values of $a, b, c$ and the polynomial $P(x)$ .
The values are $a = -4$ , $b = 14$ , and $c = -5$ .
The polynomial $P(x)$ is $x^3 - 4x^2 + 14x - 5$ .
The final answer is $\boxed{a=-4, b=14, c=-5, P(x) = x^3 - 4x^2 + 14x - 5}$ .

Find the remainder when P(x) = 3x4 − 2x3 + 5x − 7 is divided by x+1. Let P(x) = x3 + ax2 + bx + c. Find a,b,c if P(1)=6, P(2)=15, P(3)=28.

Answer

Step-by-step solution

Apply the Remainder Theorem.

Substitute $x=-1$ into $P(x)$ .

Set up a system of equations using the given conditions.

Solve the system of equations.

State the values of $a, b, c$ and the polynomial $P(x)$ .

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