Identify constant, linear, quadratic, cubic and quartic polynomials from the following.
(i) – 7 + x
(ii) 6y
(iii) – ?^?
(iv) 1 – y – ?^?
(v) x – ?^? + ?^?
(vi) 1 + x + ?^?
(vii) -6?^?
(viii) -13
(ix) –p
(i) Give an example of a monomial of degree 5.
(ii) Give an example of a binomial of degree 8.
(iii) Give an example of a trinomial of degree 4.
(iv) Give an example of a monomial of degree 0.
Rewrite each of the following polynomials in standard form.
(i) x – 2?^? + 8 + 5?^?
(ii) ?/? + 4?^?- 3y + 2?^?
(iii) ??^? + 2x – ?^?- 3?^?
(iv) ? + ? − ??^?+ ?^? – ?^?
If p(x) = 5 – 4x + 2?^?, find
(i) p(0)
(ii) p(3)
(iii) p(-2)
If p(y) = 4 + 3y – ?^?+ 5?^?, find
(ii) p(2)
(iii) p(-1)
If f(t) = 4?^? – 3t + 6, find
(i) f(0)
(ii) f(4)
(iii) f(-5)
If p(x) = ?^?- 3?^?+ 2x, find p (0), p (1), p (2). What do you conclude?
If p(x) = ?^?+ ?^?- 9x – 9, find p (0), p (3), p (-3) and p (-1). What do you conclude about the zeros of p(x)? Is 0 a zero of p(x)?
Verify that
(i) 4 is a zero of the polynomial, p(x) = x – 4.
(ii) -3 is a zero of the polynomial, q(x) = x + 3.
(iii)?/? is a zero of the polynomial, f(x) = 2-5x.
(iv)−?/? is a zero of the polynomial, g(y) = 2y+1.
(i) 1 and 2 are the zeros of the polynomial, p(x) = ?^?- 3x + 2.
(ii) 2 and -3 are the zeros of the polynomial, q(x) = ?^?+x -6.
(iii) 0 and 3 are the zeros of the polynomial, r(x) = ?^?- 3x.
Find the zero of the polynomial:
(i) p(x) = x-5
(ii) q(x) = x+4
(iii) r(x) = 2x+5
(iv) f(x) = 3x+1
(v) g(x) = 5 – 4x
(vi) h(x) = 6x – 2
(vii) p(x) = ax, a ≠0
(viii) q(x) = 4x
If 2 and 0 are the zeros of the polynomial f(x) = 2?^?- 5?^?+ ax + b then find the values of ‘a’ and ‘b’.
By actual division, find the quotient and the remainder when (?^? + ?) is divided by (x-1). Verify that remainder = f(1).
Verify the division algorithm for the polynomials
p(x) = 2?^? − ??^? + ??^? − ? + ? and g(x) = x+2
Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where
p(x) = ??^? − ??^? + ?? − ??, g(x) = x-3
p(x) = ??^? − ??^? − ?? − ?, g(x) = x-2
p(x) = 2?^? − ??^? + ? + ??, g(x) = 2x -3
p(x) = ?^? − ??^? − ?? − ?, g(x) = x+1
p(x) = ??^? + ?^? − ??? − ??, g(x) = x+2
p(x) = ??^? + ???^? + ?, g(x) = 3x+2
p(x) = ?^? − ??^? + ?? − ?, g(x) = 1 – ?/?x
p(x) = ??^? + ??^? − ??? − ?, g(x) = (x+?/?)
p(x) = ?^? − ??^? + ?? − ?, g(x) = x-a
The polynomials (??^? + ?^? − ?? + ?) and (??^?-3?^?-3x+a) when divided by (x-2) leave the same remainder. Find the value of a.
The polynomial p(x) = ?^? − ??^? + ??^? − ?? + ? when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively. Find the values of ‘a’ and ‘b’. Hence find the remainder when p(x) is divided by (x-2).
If p(x) = ?^? − ??^? + ?? − ? and g(x) = x-2, show that p(x) is not a multiple of g(x).
If p(x) = ??^? − ???^? − ?? + ? and g(x) = 2x+1, show that g(x) is not a factor of p(x).
Using factor theorem, show that g(x) is a factor of p(x), when
p(x) = ??^? + ??^? − ??? − ??, g(x) = x-3
p(x) = ??^? + ??^? + ??^? − ??? − ?, g(x) = x-1
p(x) = ?^? − ?^? − ??, g(x) = x + 2
p(x) = 69 + 11x – ?^? + ?^?, g(x) = x+3
p(x) = ??^? + ??^? − ??? − ??, g(x) = x+5
p(x) = ??^? + ?^? − ??^? − ? + ?, g(x) = 2x – 3
p(x) = ??^? + ?^? − ??? + ??, g(x) = 3x – 2
p(x) = ??^? − ?√?? − ?, g(x) = x – √?
p(x) = ?√??^? + ?? + √?, g(x) = x+√?
Show that (p-1) is a factor of (?^?? − ?) and also of (?^?? − ?).
Find the value of k for which (x-1) is a factor of (??^? + ??^? + ? + ?).
Find the value of a for which (x-4) is a factor of (??^? − ??^? − ??? + ?).
Find the value of a for which (x+1) is a factor of (a?^? + ?^? − ?? + ?? − ?).
Find the value of a for which (x+2a) is a factor of (?^? − ??^??^? + ?? + ?? + ?).
Find the value of m for which (2x-1) is a factor of (8?^? + ??^? − ???^? + ??? + ?).
Find the value of ‘a’ for which the polynomial (?^? − ?^? − ???^? − ? + ?)is divisible by (x+3).
Without actual division, show that (?^? − ??^? − ??? + ??)is exactly divisible by (?^?+2x-3).
If (?^? + ??^? + ?? + ?) has (x-2) as a factor and leaves a remainder 3 when divided by (x-3), find the values of ‘a’ and ‘b’.
Find the values of ‘a’ and ‘b’ so that the polynomial (?^? − ???^? + ?? + ?) is exactly divisible by (x-1) as well as (x-2).
Find the values of a and b so that the polynomial (?^? + ??^? − ??^? − ?? + ?) is exactly divisible by (x+2) as well as(x+3).
If both (x-2) and (x – ?/?) are factors of p?^?+ 5x + r, prove that p = r.
Without actual division, prove that ??^? − ??^? + ??^? − ? + ? is divisible by ?^?- 3x + 2.
What must be added to ??^? − ??^? + ??^? − ? − ? so that the result is exactly divisible by (x-2)?
What must be subtracted from (?^? + ??^? − ??^? + ?? + ?) so that the result is exactly divisible by
(?^?+2x – 3)?
Use factor theorem to prove that (x + a) is a factor of (?^? + ?^?)for any odd positive integer n.
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