If alpha and beta are zeroes if the polynomial 2x²-7x+5 then find a polynomial whose zeroes are 2 alpha + 1 and 2 beta+ 3 ??

Answer 1

# x^2-11x+30#.

Let, the poly. be #p(x)=2x^2-7x+5#.
Then, #p(x)=0 rArr 2x^2-7x+5=0#.
#:. ul(2x^2-5x)-ul(2x+5)=0#.
#:. x(2x-5)-1(2x-5)=0#.
#:. (2x-5)(x-1)=0#.
#:." The zeroes of "p(x)" are "5/2, and, 1#.
We select, #alpha=5/2 and beta=1#.
Letting #alpha_0=2alpha+1, and beta_0=2beta+3#, we have,
#alpha_0=2(5/2)+1=6, and beta_0=2(1)+3=5#.
#:. alpha_0+beta_0=6+5=11, and alpha_0*beta_0=6*5=30#.
So, the desired poly. #P(x)# with these zeroes is given by,
#P(x)=x^2-(alpha_0+beta_0)x+alpha_0*beta_0, i.e., #
#P(x)=x^2-11x+30#.
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Answer 2

If α and β are the zeros of the polynomial 2x² - 7x + 5, then by Vieta's formulas, their sum is (-b)/a and their product is c/a.

Given: α + β = 7/2 αβ = 5/2

Now, if the zeros of a polynomial are (2α + 1) and (2β + 3), then by using the sum and product of roots, we can find the polynomial.

The sum of the new roots: (2α + 1) + (2β + 3) = 2(α + β) + 4

The product of the new roots: (2α + 1)(2β + 3) = 2(αβ) + 6(α + β) + 3

Now, we can use these to form the polynomial.

The polynomial is: [x - (2α + 1)][x - (2β + 3)] = 0

Expanding this expression will give the polynomial.

Polynomial: (x - (2α + 1))(x - (2β + 3)) = 0

Expanding: x² - (2α + 2β + 4)x + (2α + 1)(2β + 3)

Now, we replace (2α + 2β + 4) with 7 (from the sum of roots) and (2α + 1)(2β + 3) with 5/2 (from the product of roots):

x² - 7x + 5/2

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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