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If both x – 2 and x – 1/2 are factors of px² + 5x + r, show that p = r.

Given f(x) = px2+5x+r and factors are x-2, x – 1/2 To Prove We have to prove that p = r.. Proof f(x) = px2+5x+r and... View Article

The polynomial p(x) = x⁴ – 2x³ + 3x² – ax + 3a – 7 when divided by x + 1 leaves the remainder 19. Find the values of a. Also find the remainder when p(x) is divided by x + 2.

Given p(x) = x4 – 2x3 + 3x2 – ax + 3a – 7 when divided by x + 1 leaves the... View Article

If the polynomials az³ + 4z² + 3z – 4 and z³ – 4z + a leave the same remainder when divided by z – 3, find the value of a.

Given p(z)=az3 + 4z2 + 3z – 4 p1(z)= z3 – 4z + a on dividing by z – 3 leave the same reminder Find... View Article

If + 2 is a factor of ⁵ – 4 2 ³ + 2 + 2 + 3, find a.

Solution According to the given details Let p(x) = x5 – 4a2x3 + 2x + 2a + 3 g(x) = x + 2a g(x) = 0... View Article

For what value of m is ³ – 2 ² + 16 divisible by x + 2?

Solution According to the given details Let p(x) = x 3 – 2mx2 + 16, g(x) = x + 2 g(x) = 0 ⟹ x + 2... View Article

Show that p – 1 is a factor of p¹⁰ – 1 and also of p¹¹ – 1.

Solution Let f(p) = 10 − 1,and m(p) = – 1 zero of m(p) ⇒ m(p) = 0 p – 1 = 0 p = 1... View Article

Determine which of the following polynomials has x – 2 a factor: (i) 3 ² + 6 −24. (ii) 4 ² + −2.

Solution (i) 3 2 + 6 −24. Let p(x) =3 2 + 6 −24 g(x) = x – 2 g(x) = x – 2 zero of g(x) ⇒ g(x) =... View Article

Show that: (i) + 3 is a factor of 69 + 11 − ² + ³. (ii) 2 −3 is a factor of + 2 ³ – 9 ² + 12

Solution (i) + 3 is a factor of 69 + 11 − 2 + 3. Let us assume that p(x) = 69 + 11x −... View Article

Check whether p( ) is a multiple of g( ) or not: (i) p( ) = ³ – 5 ² + 4 – 3, g( ) = – 2 (ii) p( ) = 2 ³ – 11 ²− 4 + 5, ( ) = 2 + 1

Solution (i) p( ) = 3 – 5 2 + 4 – 3, g( ) = – 2 According to the given details g(x)=x –... View Article

By Remainder Theorem find the remainder, when p(x) is divided by g(x), where (i) p( ) = ³ – 2 ² – 4 – 1, g( ) = + 1 (ii) p( ) = ³ – 3 ² + 4 + 50, g( ) = – 3 (iii) p( ) = 4 ²³ – 12 ² + 14 – 3, g( ) = 2 – 1 (iv) p( ) = ³ – 6 ² + 2 – 4, g( ) = 1 – 3/2

Solution The remainder theorem says that if a polynomial p(x) is divided by a linear factor (x-a) then the remainder is obtained as p (a) (i)... View Article

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: x⁴ + 1; x -1

Solution On performing the long division method to the given polynomial x4 + 1 by x-1 we obtain Hence, Quotient = x3 +... View Article

Find the zeroes of the polynomial: p( ) = ( -2)²−( + 2)²

Solution p(x) = ( –2)2−( + 2)2 Zero of the polynomial p(x) = 0 Hence, ⇒ (x–2)2−(x +... View Article

Find the zeroes of the polynomial in each of the following: (i) p(x) = x – 4 (ii) g(x) = 3 – 6x (iii) q(x) = 2x -7 (iv) h(y) = 2y

Solution (i) p(x) = x – 4 Zero of the polynomial p(x) ⇒ p(x) = 0 P(x) = 0 ⇒ x – 4= 0 ⇒ x = 4 Thus, the... View Article

Verify whether the following are true or false: (i) -3 is a zero of x – 3 (ii) – 1/3 is a zero of 3x + 1 (iii) – 4/5 is a zero of 4 -5y (iv) 0 and 2 are the zeroes of t² – 2t (v) -3 is a zero of y² + y – 6

Solution (i) –3 is a zero of x – 3 The given statement is False Zero of x – 3 can be determined as x – 3 = 0... View Article

Find p(0), p(1), (−2) for the following polynomials: (i) ( ) = 10 −4 ² -3 (ii) ( ) = (y + 2) (y – 2)

Solution (i) ( )=10 −4 2 –3 p(x) = 10 −4 2 –3 When x = 0, p(x) = p(0) On substituting x = 0, p(0) =... View Article

If p( ) = ² – 4 + 3, evaluate: (2)− (−1) + (1/2).

Given p( ) = 2 – 4 + 3 According to the given details When x = 2, p(x) = p(2) p( ) = 2 – 4 + 3... View Article

Find the value of the polynomial 3 ³ – 4 ² + 7 – 5, when x = 3 and also when x = -3.

Given p(x) = 3 3 – 4 2 + 7 – 5 According to the given details When x = 3, p(x) = p(3) p(x) = 3... View Article

Give an example of a polynomial, which is: (i) monomial of degree 1 (ii) binomial of degree 20 (iii) trinomial of degree 2

Solution (i) monomial of degree 1 A monomial is an algebraic expression that has only one term. An example of a polynomial, which is a... View Article

Classify the following as a constant, linear, quadratic and cubic polynomials: (i) 2 – x² + x³ (ii) 3x³ (iii) 5t – √7 (iv) 4 – 5y² (v) 3 (vi) 2 + x (vii) y³ – y (viii) 1 + x + x² (ix) t² (x) √2x – 1

Solution A degree of a polynomial is the highest or the greatest power of a variable in a polynomial equation. The degree indicates The... View Article

Write the coefficient of x²in each of the following: (i) (π/6)x + x² – 1 (ii) 3x – 5 (iii) (x -1) (3x – 4) (iv) (2x – 5) (2x² – 3x + 1)

Solution A coefficient is an integer that is multiplied to the variable of a single term or the terms of a given polynomial. ... View Article

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