If alpha and beta are zeroes of x^2+5x+6, what is the new polynomial whose zeroes are alpha+1 and beta+1?
Answer and explanation
Correct answer: x^2+3x+2
The governing idea is transformation of zeroes. Factor the original polynomial: x^2+5x+6=(x+2)(x+3), so alpha=-2 and beta=-3. After adding 1, the new zeroes are -1 and -2. A monic quadratic with these zeroes is (x+1)(x+2)=x^2+3x+2, so option A is correct. The same result follows from sums and products: the new sum is (alpha+1)+(beta+1)=-5+2=-3, so the x-coefficient is 3; the new product is (-1)(-2)=2. Option B is the original polynomial, option C shifts the roots in the wrong direction, and option D has the wrong sign for the x-term.
Frequently asked questions
What is the correct answer to this question?
x^2+3x+2
Why is this the correct answer?
The governing idea is transformation of zeroes. Factor the original polynomial: x^2+5x+6=(x+2)(x+3), so alpha=-2 and beta=-3. After adding 1, the new zeroes are -1 and -2. A monic quadratic with these zeroes is (x+1)(x+2)=x^2+3x+2, so option A is correct. The same result follows from sums and products: the new sum is (alpha+1)+(beta+1)=-5+2=-3, so the x-coefficient is 3; the new product is (-1)(-2)=2. Option B is the original polynomial, option C shifts the roots in the wrong direction, and option D has the wrong sign for the x-term.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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