Question

find the zeros of the polynomial P(x)=(x+4)(x+9) and verify the relationship between the zeros and coefficients​

Answer 1

We have,

p(x)=(x+4)(x+9)

Let @ and ß be the zeros of the above polynomial

By Remainder Theorm,

p(x)=0

→(x+4)(x+9)=0

→x= -4 or x= -9

→@= -4 and ß= -9

Now,

Sum of Zeros:

@+ß

= -4+(-9)

= -13

Product of zeros:

@ß = (-4)(-9)=36

Hence, verified

Answer 2

Q4. If two zeroes of the polynomial x4 6x3 – 26x2 + 138x – 35 are find other zeroes.

Sol. Here, p(x) = x4 – 6x3 – 26x2 + 138x – 35.

∴Two of the zeroes of p (x) are:

Now, dividing p (x) by x2 – 4x + 1, we have:

∴(x2 – 4x + 1)(x2 – 2x – 35) = p(x)

⇒ (x2 – 4x + 1) (x – 7) (x + 5) = p(x)

i.e., (x – 7) and (x + 5) are other factors of p(x).

∴7 and –5 are other zeroes of the given polynomial.

Q5. If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be (x + α ), find k and α.

Sol. Applying the division algorithm to the polynomials x4 – 6x3 + 16x2 – 25x + 10 and x2 – 2x + k, we have:

∴Remainder = (2k – 9) x – k(8 – k) + 10

But the remainder = x + α

Therefore, coparing them, we have:

and

α = –k(8 – k) + 10

= –5(8 – 5) + 10

= –5(3) + 10

= –15 + 10

= –5

Thus, k = 5 and α = –5