When the polynomial $f(x)$ is divided by $x - 3,$ the remainder is 15. When $f(x)$ is divided by $(x - 1)^2,$ the remainder is $
1 answer:
The remainder when f(x) is divided by (x - 3)(x - 1)^2 is (6x - 3).
<h3>What is the remainder theorem for polynomials?</h3>If there is a polynomial p(x), and a constant number 'a', then
where g(x) is a factor of p(x)
Here, f(x) is the given polynomial.
By Remainder Theorem,
When divided by (x-3),
f(3) = 15........(1)
When divided by (x-1)²,
f(1) = 2x - 1........(2)
Another polynomial is (x - 3)(x - 1)²
= (x - 3)(x - 1)(x -1)
So,
f(x) = (x - 3)(x - 1)(x -1)Qx + (ax+b)
For f(1),
2x + 1 = a + b
3 = a + b
For f(3),
15 = 3a + b
or, 15 = 3a + 3 - a
or, 15 = 3 + 2a
or a = 6
Also, b = -3
Hence, the remainder is (6x - 3).
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The width of the rectangle is 11 inches.
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.. x = y +2
.. xy = 3(2(x +y)) -1
This can be solved a variety of ways. I find a graphing calculator provides an easy solution: (x, y) = (13, 11).
The length of the rectangle is 13 inches.
The width of the rectangle is 11 inches.
______
Just so you're aware, the problem statement is nonsensical. You cannot compare perimeter (inches) to area (square inches). You can compare their numerical values, but the units are different, so there is no direct comparison.