Each divisor was divided into another polynomial , resulting in the given quotient and remainder. Find the other polynomial the dividend
Divisor: x+10,quotient,x^2-6x+10,remainder :-1
I am so confused help is appreciated
Well, it seems like you're dealing with polynomial division, but let's try to lighten the mood a bit with a joke before we dive into the math:
Why did the polynomial go to the shrink?
Because it had too many imaginary friends!
Okay, now let's get back to business. To find the dividend, we can use the information given: the divisor, the quotient, and the remainder. We know the divisor is x + 10, the quotient is x^2 - 6x + 10, and the remainder is -1.
In polynomial division, the dividend is equal to the product of the divisor and the quotient plus the remainder. So, we can write the equation as:
Dividend = Divisor * Quotient + Remainder
Now, substitute the values we have:
Dividend = (x + 10) * (x^2 - 6x + 10) - 1
Multiplying (x + 10) by (x^2 - 6x + 10) through multiplication of each term, we get:
Dividend = x^3 - 6x^2 + 10x + 10x^2 - 60x + 100 - 1
Simplifying further, we have:
Dividend = x^3 + 4x^2 - 50x + 99
And that's it! The other polynomial, the dividend, is x^3 + 4x^2 - 50x + 99.
Remember, if you need more help or have any other questions, don't hesitate to ask. And if you need a good laugh, I'm here with more jokes!
To find the dividend, we can use the division algorithm for polynomials.
The division algorithm states that if we have a polynomial divident f(x) and a polynomial divisor g(x), then there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that:
f(x) = g(x) * q(x) + r(x)
Given:
Divisor: x + 10
Quotient: x^2 - 6x + 10
Remainder: -1
Let's substitute the given values into the division algorithm:
f(x) = (x + 10)(x^2 - 6x + 10) + (-1)
Now, we need to multiply the divisor by the quotient:
f(x) = (x + 10)(x^2 - 6x + 10) - 1
Using the distributive property, we can expand the expression:
f(x) = x(x^2 - 6x + 10) + 10(x^2 - 6x + 10) - 1
Multiplying each term by the corresponding term in the parenthesis:
f(x) = x^3 - 6x^2 + 10x + 10x^2 - 60x + 100 - 1
Combining like terms:
f(x) = x^3 + 4x^2 - 50x + 99
Therefore, the polynomial dividend is f(x) = x^3 + 4x^2 - 50x + 99.
To find the other polynomial, we will use the polynomial division process.
Step 1: Write the divisor, dividend, quotient, and remainder in the following format:
Divisor = Quotient * Dividend + Remainder
Step 2: Replace the divisor, quotient, and remainder with their respective values:
(x + 10) = (quotient) * (dividend) + (-1)
Step 3: Rearrange the equation to solve for the dividend:
(x + 10) + 1 = (quotient) * (dividend)
Step 4: Simplify the equation:
x + 11 = (quotient) * (dividend)
Now, we need to find the value of the dividend.
Step 5: Expand the dividend term (quotient) * (dividend):
x + 11 = (quotient) * (dividend)
x + 11 = (x^2 - 6x + 10) * (dividend)
Step 6: Since the degree of the divisor (x + 10) is 1, the degree of the dividend should be 2 in order for the equation to be balanced. Therefore, the dividend must be a quadratic polynomial.
Step 7: Multiply the divisor (x + 10) by the quadratic polynomial (dividend) and set it equal to x + 11:
x + 11 = (x^2 - 6x + 10)(dividend)
Step 8: Expand the right side of the equation:
x + 11 = x^3 - 6x^2 + 10x - 6x^2 + 36x - 60 + 10(dividend)
Step 9: Simplify:
x + 11 = x^3 - 12x^2 + 46x - 60 + 10(dividend)
Step 10: Combine like terms:
x + 11 = x^3 - 12x^2 + 46x - 50 + 10(dividend)
Step 11: Set the coefficients of the like terms equal to each other:
x = x^3 - 12x^2 + 46x - 50
Step 12: Rearrange the equation to the standard form:
x^3 - 12x^2 + 45x - 50 - x = 0
Step 13: Combine like terms:
x^3 - 12x^2 + 45x - 50 - x = 0
x^3 - 12x^2 + 44x - 50 = 0
Therefore, the other polynomial, which is the dividend, is:
x^3 - 12x^2 + 44x - 50
I think what you're being asked is this. There's some polynomial - let's call it f(x) - that when divided by (x + 10) equals (x^2 - 6x + 10) with remainder -1. What is f(x)?
If I'm right, then we should be able to find it by multiplying (x^2 - 6x + 10) by (x + 10), and then adding -1 to the result. Let's do that:
To multiply (x^2 - 6x + 10) by (x + 10), multiply (x^2 - 6x + 10) by x, then multiply (x^2 - 6x + 10) by 10, and add the two together. That is:
The product of (x^2 - 6x + 10) and x is (x^3 - 6x^2 + 10x).
The product of (x^2 - 6x + 10) and 10 is (10x^2 - 60x + 100).
The sum of the above two expressions is (x^3 + 4x^2 - 50x + 100). Now add that -1 to it to allow for the remainder you were told about earlier: that gives (x^3 + 4x^2 - 50x + 99).
I think that's the answer, so now let's check it. If I'm right, then (x^3 + 4x^2 - 50x + 99) divided by (x + 10) should equal (x^2 - 6x + 10) with remainder -1. Does it?
To divide (x + 10) into (x^3 + 4x^2 - 50x + 99), ask what is x^3 divided by x? The answer is x^2, so that's the first (squared) term of my quotient. Next, multiply (x + 10) by x^2 and subtract the result from (x^3 + 4x^2 - 50x + 99), just as you would when doing the first step of a long division sum:
(x^3 + 4x^2 - 50x + 99) minus (x^3 + 10x^2) equals (-6x^2 - 50x + 99).
To divide (-6x^2 - 50x + 99) by (x + 10), ask what is -6x^2 divided by x? The answer is -6x, which is the second (linear) term of my quotient. Next, multiply (x + 10) by -6x and subtract the result from (-6x^2 - 50x + 99), again just as you would when doing the second step of a long division sum:
(-6x^2 - 50x + 99) minus (-6x^2 - 60x) equals (10x + 99).
Finally, (10x + 99) is 10 times (x + 10) remainder -1. So 10 is the third (constant) term of my quotient. So the complete answer is the first (squared) term of my quotient plus the second (linear) term of my quotient plus the third (constant) term of my quotient, which is (x^2 - 6x + 10) remainder -1. So I think I've got it right.
Does that help? It would be easier to express the above if I could write it out like a proper long division sum, but I'm hoping you can relate the above to the examples you'll have seen in your class, which will probably have been written out this way.