1)when a certain polynomial is divided by x - 3, the quotient is x^2+2x-5 and the remainder is -3. what is the polynomial
2) Find each quotient and remainder. Assume the divisor is not equal to zero.
a)(2x^2+29x-x^3-40)/(-3+x)
b)(6+7x-11x^2-2x^3)/(x+9)
c)(x^3-2x^2+4x+150/(x^2+2x-3)
d)(3x^3+2x^2-11x-12)/(x+1)
e)(x^2+x^2y-9xy^2-9y^3)/(x+y)
1)
P = Your polynome
P / ( x -3 ) = x ^ 2 + 2 x - 5 - 3 / ( x - 3 ) Multiply both sides by ( x - 3)
P = ( x ^ 2 + 2 x - 5 ) * ( x - 3 ) - 3 * ( x - 3 ) / ( x - 3 )
P = x * x ^ 2 + x * 2 x - 5 * x - 3 * x ^ 2 - 3 * 2 x - ( - 3 ) * 5 - 3 * 1
P = x ^ 3 + 2 x ^ 2 - 5 x - 3 x ^ 2 - 6 x + 15 - 3
P = x ^ 3 - x ^ 2 - 11 x + 12
2 )
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1) The polynomial is (x - 3)(x^2 + 2x - 5) - 3.
2)
a) Quotient: 3x^2 + 23x + 66, Remainder: -182
b) Quotient: -2x^2 + 49x - 442, Remainder: 3080
c) Quotient: x - 4, Remainder: 174
d) Quotient: 3x^2 - x - 12, Remainder: 0
e) Quotient: x - 10y^2, Remainder: -9y^3
1) To find the polynomial, we can use the remainder theorem. The remainder theorem states that if a polynomial f(x) is divided by x - a, the remainder is f(a). In this case, we are given that the remainder is -3 when the polynomial is divided by x - 3.
Using the remainder theorem, we substitute x = 3 into the quotient to find the remainder polynomial:
R(x) = x^2 + 2x - 5
R(3) = (3)^2 + 2(3) - 5
R(3) = 9 + 6 - 5
R(3) = 10
Therefore, the polynomial is x^2 + 2x - 5 with a remainder of -3.
2) Let's solve each of the given division problems:
a) (2x^2 + 29x - x^3 - 40) / (-3 + x)
Quotient: -x^2 + 30
Remainder: -110
b) (6 + 7x - 11x^2 - 2x^3) / (x + 9)
Quotient: -2x^2 - 2x + 1
Remainder: -63
c) (x^3 - 2x^2 + 4x + 150) / (x^2 + 2x - 3)
Quotient: x - 4
Remainder: 162x + 642
d) (3x^3 + 2x^2 - 11x - 12) / (x + 1)
Quotient: 3x^2 - x - 12
Remainder: 0
e) (x^2 + x^2y - 9xy^2 - 9y^3) / (x + y)
Quotient: x - y
Remainder: -8y^2
These are the solutions for each of the division problems given.
1) To find the polynomial, we will perform polynomial long division.
Step 1: Write the dividend (the polynomial being divided) and divisor:
Dividend: Let's call it P(x)
Divisor: (x - 3)
Step 2: Divide the first term of the dividend (in this case, x^2) by the first term of the divisor (which is x), which gives us x.
Write this as the first term in the quotient.
Step 3: Multiply the divisor by x (the term we just obtained) and subtract it from the dividend:
(x - 3) * x = x^2 - 3x
P(x) - (x^2 - 3x) = 2x - 5
Step 4: Bring down the next term from the dividend, which is 2x:
Dividend: 2x - 5
Step 5: Repeat steps 2 to 4 until there are no more terms in the dividend.
Divide (2x - 5) by (x - 3):
Step 2: (2x) / (x) = 2
Write 2 as the next term of the quotient.
Step 3: Multiply (x - 3) by 2 and subtract it from (2x - 5):
(2)(x - 3) = 2x - 6
(2x - 5) - (2x - 6) = 1
Step 4: There are no more terms to bring down.
The quotient is x^2 + 2x - 5, and the remainder is 1. Therefore, the original polynomial is:
(x - 3)(x^2 + 2x - 5) + 1.
2) Let's go through each division one by one:
a) (2x^2 + 29x - x^3 - 40) / (-3 + x)
To find the quotient and remainder, divide (2x^2 + 29x - x^3 - 40) by (-3 + x).
b) (6 + 7x - 11x^2 - 2x^3) / (x + 9)
To find the quotient and remainder, divide (6 + 7x - 11x^2 - 2x^3) by (x + 9).
c) (x^3 - 2x^2 + 4x + 150) / (x^2 + 2x - 3)
To find the quotient and remainder, divide (x^3 - 2x^2 + 4x + 150) by (x^2 + 2x - 3).
d) (3x^3 + 2x^2 - 11x - 12) / (x + 1)
To find the quotient and remainder, divide (3x^3 + 2x^2 - 11x - 12) by (x + 1).
e) (x^2 + x^2y - 9xy^2 - 9y^3) / (x + y)
To find the quotient and remainder, divide (x^2 + x^2y - 9xy^2 - 9y^3) by (x + y).