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Sir, How do i solve polynomial-linear word problem effectively. For example, Find the values of the constants p,q and r such that,when the polynomial f(x)=x^3+px^2+qx+r is divided by(x+2),(x-1) and (x-3),the remainders are, respectively, -48, 0 and 2. Thuse,factorize f(x) completely.

Hi Johnny,

When the polynomial f(x) is divided by x-a, the remainder is given by f(a).

For the polynomial f(x) = x³ + px² + qx + r, and from the information provided, we have

f(-2) = (-2)³ + p(-2)² + q(-2) + r = -48

-8 + 4p - 2q + r = -48

4p - 2q + r = -40

f(1) = (1)³ + p(1)² + q(1) + r = 0

1 + p + q + r = 0

p + q + r = -1

f(3) = (3)³ + p(3)² + q(3) + r = 2

27 + 9p + 3q + r = 2

9p + 3q + r = -25

and the three equations solved simultaneously to get

p = -5, q = 8, r = -4

So,

f(x) = x³ - 5x² + 8x - 4

As it is the case that dividing by x-1 gives a remainder of 0, we conclude that x-1 is a factor. To factorize f(x) completely, we first divide by x-1 and then factorize the resulting quadratic expression.

Now,

(x³ - 5x² + 8x - 4) / (x-1) = x² - 4x + 4

= (x-2)²

Therefore,

f(x) = x³ - 5x² + 8x - 4 = (x-1)(x-2)²

Regards

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Comment | Hurrah!!! the great mathematics Ace(Ahmed salami),if possible to increase your knowledgeability to 50(more than highest) i will do it. because, you shower me with a whoops of delight by help me ellucedating and solving "polynomial_word_problem" which seems to be more complicated topic for me in mathematics. infact, you are league of your own(much better than others) in mathematics. Thanks. |

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