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Reptile [31]
3 years ago
7

The polynomial p{x) = x4 -2x3 + 3x2 -ax+3a-7 when divided by x+1 leaves the remainder 19. Find the values of a. Also, find the r

emainder when p(x) is divided by x+ 2.
Mathematics
1 answer:
Mrrafil [7]3 years ago
5 0

Answer:

a = 5

Remainder when p(x) is divided by x+2 = 62

Step-by-step explanation:

Given:

P(x) = x⁴-2x³+3x²-ax+3a-7

When x+1 divides the polynomial p(x) the ramainder is 19.

Applying remainder theorem,

x = -1

p(-1) = 19

Substitute the x = -1 into the polynomial expression

p(-1) = (-1)⁴-2(-1)³+3(-1)²-a(-1)+3a-7 = 19

1+2+3+a+3a-7 = 19

6-7+4a = 19

4a-1 = 19

4a = 19+1

4a = 20

a = 20/4

a = 5.

Hence, a = 5

p(x) = x⁴-2x³+3x²-5x+8

If p(x) is divided by x+2,

Then the remainder is p(-2)

p(-2) = (-2)⁴-2(-2)³+3(-2)²-5(-2)+8

p(-2) = 16+16+12+10+8

p(-2) = 62

Hence the remaider when p(x) is divided by x+2 is 62

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Step-by-step explanation:

<u>Step 1: Define</u>

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<u>Step 2: Evaluate</u>

  1. Substitute in <em>x</em> [Function f(x)]:                                                                           f(-3) = 2(-3)²
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==========================================================

Explanations:

Part (a)

The set of natural numbers is {1,2,3,4,...} which is the set of positive whole numbers. We don't include 0. Note that \sqrt{64} = 8 to show that it's part of the set of natural numbers. Something like -2 is not in the set, same for -2/7, etc.

------------------------------

Part (b)

The set of whole numbers is {0,1,2,3,...} which is almost identical to the previous set, but we're now including 0.

------------------------------

Part (c)

The set of integers is {..., -3, -2, -1, 0, 1, 2, 3, ...}

We are including the negative values as well as the positive values too. Zero is included.

------------------------------

Part (d)

Any rational number is of the form p/q, where p,q are integers and q is nonzero. Something like -2/7 is rational. Here we have p = -2 and q = 7.

Any whole number is rational. Eg: 8 = 8/1

Any decimal that terminates is rational. The value 7.1 is the same as 71/10

------------------------------

Part (e)

If a number isn't rational, then it's considered irrational.

Something like \sqrt{7} is irrational because 7 isn't a perfect square.

The decimal version of the number \sqrt{7} \approx 2.6457513110646 goes on forever without any pattern. So this is more evidence it's irrational.

------------------------------

Part (f)

If your teacher hasn't covered imaginary or complex numbers, then every value you encounter so far is a real number.

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