RD Sharma Test: Polynomials

If the graph of a polynomial intersects the x-axis at three points, then the number of zeroes are 3 because number of zeroes of the polynomial are the number of the coordinates of the points where its graph intersects the x-axis.

Sum of the zeroes of the polynomial  And Product of the zeroes of the polynomial

A polynomial of degree 3 is called a cubic polynomial. 

For example, x³−1, 4a³ − 100a² + a − 6, and m²n + mn² are all cubic polynomials with atmost 3 zeroes (having degree 3).

Hence, the polynomial having atmost 3 zero is a cubic polynomial.

Given: x³+x²−2x−3 = (x−2)(x²+ax+b)+5

Dividing L.H.S. by (x−2)

∴ (x-2)(x²+3x+4)+5 = (x-2)(x²+az+b)+5 
Comparing both side, we have a = 3, b = 4

If The graph of the polynomial f(x) = 2x – 5 intersects the x – axis then y = 0 
2x−5 = 0 ⇒ x = 5/2

Here   ……….(i)
And it is given that α−β = 1 ……….(ii)
On solving eq. (i) and eq. (ii),
we get

Let α,β,γ are the zeroes of the given polynomial. Given: α = 2
 Since αβγ = -d/a 

The maximum number of zeroes that a polynomial of degree 3 can have is three because the number of zeroes of a polynomical is equals to the degree of that polynomial.

Polynomial is formed composed of the phrases Nominal, which means "terms," and Poly, which means "many." When exponents, constants, and variables are combined using mathematical operations like addition, subtraction, multiplication, and division, the result is a polynomial (No division operation by a variable). The classification of the phrase as a monomial, binomial, or trinomial depends on how many terms are included in it. Here are some examples of constants, variables, and exponents:

Constants. For instance, 1, 2, 3, etc.

Variables. For instance, g, h, x, y, etc.

Exponents: For instance, "5 in x5"

Given:

 so that the polynomial  divides it exactly.

Find:

To find the real number which should be subtracted from the given polynomial.

Solution:

Hence 7 should be subtracted so that the polynomial  is exactly divisible by 

Therefore, 7 should be subtracted.

If the graph of a cubic polynomial intersects thrice x-axis, then the zeroes of the cubic polynomial are coordinates of x-axis.

According to question, p(2) = 0 
⇒ (2)²+3×2+k = 0
⇒ 4+6+k = 0
⇒ k = −10

Let α,β,γ are the zeroes of the given polynomial. Given: α+β+γ = (-3/4)

Since √2 and −√2 are the zeroes of 2x⁴−3x³−3x²+6x−2, then(x−√2) (x+√2) are the factors of given polynomial i.e., 
(x−√2) (x+√2) = (x²−2) is a factor of given polynomial. 
∴p(x) = 2x⁴−3x³−3x²+6x−2
⇒ p(x) = (x²−2)(2x²−3x+1)

⇒ p(x) = (x² - 2) [2x² - 2x - x + 1]
⇒ p (x) = (x² - 2) [2x (x -1) - 1 (x - 1)]
⇒ p (x) = (x² - 2) (x -1) (2x -1)
∴ Other zeroes are x - 1 = 0 and 2x - 1 = 0 ⇒ x = 1 and x = 1/2

Let one zero of the given polynomial be athen the other zero be 
⇒ k + 4 = 3k ⇒ 2k = 4 ⇒ k = 2

According to the question, p (2) = 3x² + mx - 14 = 0
⇒ 3(2)² + m x 2 - 14 = 0
⇒ 12 + 2m - 14 = 0 ⇒ m = 1
 Also p(2) = 2x³ + nx² + x - 2 = 0

Let the zeroes be k and (1/k)
Given quadratic polynomial is p(x) = (a² + 9)x² + 45x + 6a
Recall the sum of zeroes (k + 1/k) = –45/(a² + 9)
Product of zeroes [k × (1/k)] = 6a/(a² + 9)
⇒ 6a/(a² + 9) = 1
⇒ (a² + 9) = 6a
⇒ a² – 6a + 9 = 0
⇒ a² – 2(a)(3) + 3² = 0
⇒ (a – 3)² = 0
⇒ (a – 3) = 0
∴ a = 3

Now (x²+2x−3)−(x²+x−1) = x−2
Therefore, (x−2) is the polynomial which to be added to the given polynomial.

Let one zeroes of the given polynomial be α and β. According to the question, 

Sum of the zeroes = -b/a = 0

 ⇒ -(p-3)/2 = 0
 ⇒ - (p-3) = 0×2

 ⇒ - (p-3) = 0

 ⇒ p = - 3

Given: 

On comparing, we get,a = 3,b = −3√2–,c = 1
Putting these values in general form of a quadratic polynomial ax²+bx+c, we have 3x²−3√2x+1

Let α,β are the zeroes of the given polynomial.
Given: 
 ⇒ −(−27) = 3k² ⇒ k² = 0 ⇒ k= ±3

The term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms.It involves the second and no higher power of an unknown quantity or variable.