Use the remainder theorem to find the remainder when p(x) = x^4 – 9x^3 – 5x^2 – 3x + 4 is divided by x + 3.

Question :

(1) Use the Remainder Theorem to find the remainder when P(x) = x^4 – 9x^3 – 5x^2 – 3x + 4 is divided by x + 3.

Answers & Step-by-step explanation:

(1) P(x) = (x + 3)Q(x) + R  

P(-3) = (0)Q(x) + R  

i P(-3) = R  

Now, P(-3) = (-3)⁴ - 9(-3)³ - 5(-3)² - 3(-3) + 4  

So  P(-3) = 81 + 243 - 45 + 9 + 4  

=. 292

Question:

(2) Use the remainder theorem to find the remainder when P(x)=x^4−9x^3−5x^2−3x+4 is divided by x+3.

The Remainder Theorem:

In mathematics, the remainder theorem states that if we divide a polynomial, p(x), by a linear polynomial, x - a, then the remainder of that division problem is equal to p evaluated at a, or p(a). We can use this theorem to determine the remainder of various division problems without having to actually perform the division.

Answer and Explanation:

When we divide p(x)=x^4−9x^3−5x^2−3x+4 by x + 3, the remainder is 292.

To find the remainder of this division problem, we can use the remainder theorem. From the remainder theorem, we have that if we divide p(x)=x^4−9x^3−5x^2−3x+4 by x + 3, the remainder will be equal to p(x) evaluated at -3, or p(-3). Thus, to find the remainder, we simply evaluate p(-3).

p(−3)=(−3)^4−9(−3)^3−5(−3)^2−3(−3)+4=81+243−45+9+4=292

We get that p(-3) = 292, so when we divide p(x) by x + 3, we get a remainder of 292.

(3) use the remainder theorem to find the remainder when P(x)=x^4-9x^3-5x^2-3x+4 is divided by x+3. show your work if you answer please.

L(-3)  = 292

Step-by-step explanation:

Plug in -3 for x

L(-3) = (-3)^4-9(-3)^3-5(-3)^2-3(-3)+4

A negative raised to an even power is as though it were a positive number being raised to that power. So replace those -3's with 3's.

3^4-9(-3)^3-5(3)^2-3(-3)+4

Use PEMDAS

81-9(-27)-5(9)+9+4

81+243-45+9+4

L(-3) = 292

(4) f(x)/(x-a) = f(a)

here a = -3

so we need f(-3)

f(-3) =(-3)^4-9(-3)^3-5(-3)^2-3(-3)+4

= 81 + 9*27 - 45 +9 + 4

= 81 + 243 - 45 + 9 + 4

=292

(5)  

292

Explanation:

To divide by (x + 3 ) you don't have to divide by (x + 3 ) . Using the

Remainder Theorem you just have to evaluate P( - 3 ).

P(−3)=(−3)^4−9(−3)^3−5(−3)^2−3(−3)+4

= 81 + 243 - 45 + 9 + 4

= 292

ie. remainder = 292

  70 square units

Step-by-step explanation:

The spreadsheet is attached.

It shows the area of the trapezoidal shape to be 70 square units.