if polynomial p(x) has a zero x=4 then what is remainder when p(x) is divided by x-4
Answers
Dear student
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The Remainder Theorem says that we can restate the polynomial in terms of the divisor, and then evaluate the polynomial at x = a. But when x = a, the factor "x – a" is just zero! Then evaluating the polynomial at x = a gives us:
p(a) = (a – a)q(a) + r(a)
= (0)q(a) + r(a)
= 0 + r(a)
= r(a)
But remember that the remainder term r(a) is just a number! So the value of the polynomial p(x) at
x = a is the same as the remainder you get when you divide that polynomial p(x) by x – a. In terms of our concrete example:
p(4) = (4 – 4)((4)2 + 4(4) + 9) + 30
= (0)(16 + 16 + 9) + 30
= 0 + 30
= 30
But you gotta think: Okay, fine; the value of the polynomial p(x) at x = a is the remainder r(a) when you divide by x – a, but who wants to do the long division each time you have to evaluate a polynomial at a given value of x?!? You're right; this would be overkill. Fortunately, that's not what they really want you to do.
When you are dividing by a linear factor, you don't "have" to use long polynomial division; instead, you can use synthetic division, which is much quicker. In our example, we would get:
completed division: bottom row: 1 4 9 30
Note that the last entry in the bottom row is 30, the remainder from the long division (as expected) and also the value of p(x) = x3 – 7x – 6 at x = 4. And that is the point of the Remainder Theorem: There is a simpler, quicker way to evaluate a polynomial p(x) at a given value of x, and this simpler way is not to evaluate p(x) at all, but to instead do the synthetic division at that same value of x.