Identify constant, linear, quadratic, cubic and quartic polynomials from the following.
(i) – 7 + x
(ii) 6y
(iii) - 𝒛^𝟑
(iv) 1 – y - 𝒚^𝟑
(v) x - 𝒙^𝟑 + 𝒙^𝟒
(vi) 1 + x + 𝒙^𝟐
(vii) -6𝒙^𝟐
(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 – 2𝐱^𝟐 + 8 + 5𝐱^𝟑
(ii) 𝟐/𝟑 + 4𝐲^𝟐- 3y + 2𝐲^𝟑
(iii) 𝟔𝐱^𝟑 + 2x - 𝐱^𝟓- 3𝐱^𝟐
(iv) 𝟐 + 𝒕 − 𝟑𝐭^𝟑+ 𝐭^𝟒 - 𝐭^𝟐
If p(x) = 5 – 4x + 2𝒙^𝟐, find
(i) p(0)
(ii) p(3)
(iii) p(-2)
If p(y) = 4 + 3y - 𝒚^𝟐+ 5𝒚^𝟑, find
(ii) p(2)
(iii) p(-1)
If f(t) = 4𝒕^𝟐 – 3t + 6, find
(i) f(0)
(ii) f(4)
(iii) f(-5)
If p(x) = 𝒙^𝟑- 3𝒙^𝟐+ 2x, find p (0), p (1), p (2). What do you conclude?
If p(x) = 𝒙^𝟑+ 𝒙^𝟐- 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)𝟐/𝟓 is a zero of the polynomial, f(x) = 2-5x.
(iv)−𝟏/𝟐 is a zero of the polynomial, g(y) = 2y+1.
(i) 1 and 2 are the zeros of the polynomial, p(x) = 𝒙^𝟐- 3x + 2.
(ii) 2 and -3 are the zeros of the polynomial, q(x) = 𝒙^𝟐+x -6.
(iii) 0 and 3 are the zeros of the polynomial, r(x) = 𝒙^𝟐- 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) = 2𝒙^𝟑- 5𝒙^𝟐+ ax + b then find the values of ‘a’ and ‘b’.
By actual division, find the quotient and the remainder when (𝒙^𝟒 + 𝟏) is divided by (x-1). Verify that remainder = f(1).
Verify the division algorithm for the polynomials
p(x) = 2𝒙^𝟒 − 𝟔𝒙^𝟑 + 𝟐𝒙^𝟐 − 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) = 𝟐𝒙^𝟑 − 𝟕𝒙^𝟐 + 𝟗𝒙 − 𝟏𝟑, g(x) = x-3
p(x) = 𝟑𝒙^𝟒 − 𝟔𝒙^𝟐 − 𝟖𝒙 − 𝟐, g(x) = x-2
p(x) = 2𝒙^𝟑 − 𝟗𝒙^𝟐 + 𝒙 + 𝟏𝟓, g(x) = 2x -3
p(x) = 𝒙^𝟑 − 𝟐𝒙^𝟐 − 𝟖𝒙 − 𝟏, g(x) = x+1
p(x) = 𝟐𝒙^𝟑 + 𝒙^𝟐 − 𝟏𝟓𝒙 − 𝟏𝟐, g(x) = x+2
p(x) = 𝟔𝒙^𝟑 + 𝟏𝟑𝒙^𝟐 + 𝟑, g(x) = 3x+2
p(x) = 𝒙^𝟑 − 𝟔𝒙^𝟐 + 𝟐𝒙 − 𝟒, g(x) = 1 - 𝟑/𝟐x
p(x) = 𝟐𝒙^𝟑 + 𝟑𝒙^𝟐 − 𝟏𝟏𝒙 − 𝟑, g(x) = (x+𝟏/𝟐)
p(x) = 𝒙^𝟑 − 𝒂𝒙^𝟐 + 𝟔𝒙 − 𝒂, g(x) = x-a
The polynomials (𝟐𝒙^𝟑 + 𝒙^𝟐 − 𝒂𝒙 + 𝟐) and (𝟐𝒙^𝟑-3𝒙^𝟐-3x+a) when divided by (x-2) leave the same remainder. Find the value of a.
The polynomial p(x) = 𝒙^𝟒 − 𝟐𝒙^𝟑 + 𝟑𝒙^𝟐 − 𝒂𝒙 + 𝒃 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) = 𝒙^𝟑 − 𝟓𝒙^𝟐 + 𝟒𝒙 − 𝟑 and g(x) = x-2, show that p(x) is not a multiple of g(x).
If p(x) = 𝟐𝒙^𝟑 − 𝟏𝟏𝒙^𝟐 − 𝟒𝒙 + 𝟓 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) = 𝟐𝒙^𝟑 + 𝟕𝒙^𝟐 − 𝟐𝟒𝒙 − 𝟒𝟓, g(x) = x-3
p(x) = 𝟐𝒙^𝟒 + 𝟗𝒙^𝟑 + 𝟔𝒙^𝟐 − 𝟏𝟏𝒙 − 𝟔, g(x) = x-1
p(x) = 𝒙^𝟒 − 𝒙^𝟐 − 𝟏𝟐, g(x) = x + 2
p(x) = 69 + 11x - 𝒙^𝟐 + 𝒙^𝟑, g(x) = x+3
p(x) = 𝟐𝒙^𝟑 + 𝟗𝒙^𝟐 − 𝟏𝟏𝒙 − 𝟑𝟎, g(x) = x+5
p(x) = 𝟐𝒙^𝟒 + 𝒙^𝟑 − 𝟖𝒙^𝟐 − 𝒙 + 𝟔, g(x) = 2x – 3
p(x) = 𝟑𝒙^𝟑 + 𝒙^𝟐 − 𝟐𝟎𝒙 + 𝟏𝟐, g(x) = 3x – 2
p(x) = 𝟕𝒙^𝟐 − 𝟒√𝟐𝒙 − 𝟔, g(x) = x - √𝟐
p(x) = 𝟐√𝟐𝒙^𝟐 + 𝟓𝒙 + √𝟐, g(x) = x+√𝟐
Show that (p-1) is a factor of (𝒑^𝟏𝟎 − 𝟏) and also of (𝒑^𝟏𝟏 − 𝟏).
Find the value of k for which (x-1) is a factor of (𝟐𝒙^𝟑 + 𝟗𝒙^𝟐 + 𝒙 + 𝒌).
Find the value of a for which (x-4) is a factor of (𝟐𝒙^𝟑 − 𝟑𝒙^𝟐 − 𝟏𝟖𝒙 + 𝒂).
Find the value of a for which (x+1) is a factor of (a𝒙^𝟑 + 𝒙^𝟐 − 𝟐𝒙 + 𝟒𝒂 − 𝟗).
Find the value of a for which (x+2a) is a factor of (𝒙^𝟓 − 𝟒𝒂^𝟐𝒙^𝟑 + 𝟐𝒙 + 𝟐𝒂 + 𝟑).
Find the value of m for which (2x-1) is a factor of (8𝒙^𝟒 + 𝟒𝒙^𝟑 − 𝟏𝟔𝒙^𝟐 + 𝟏𝟎𝒙 + 𝒎).
Find the value of ‘a’ for which the polynomial (𝒙^𝟒 − 𝒙^𝟑 − 𝟏𝟏𝒙^𝟐 − 𝒙 + 𝒂)is divisible by (x+3).
Without actual division, show that (𝒙^𝟑 − 𝟑𝒙^𝟐 − 𝟏𝟑𝒙 + 𝟏𝟓)is exactly divisible by (𝒙^𝟐+2x-3).
If (𝒙^𝟑 + 𝒂𝒙^𝟐 + 𝒃𝒙 + 𝟔) 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 (𝒙^𝟑 − 𝟏𝟎𝒙^𝟐 + 𝒂𝒙 + 𝒃) is exactly divisible by (x-1) as well as (x-2).
Find the values of a and b so that the polynomial (𝒙^𝟒 + 𝒂𝒙^𝟑 − 𝟕𝒙^𝟐 − 𝟖𝒙 + 𝒃) is exactly divisible by (x+2) as well as(x+3).
If both (x-2) and (x - 𝟏/𝟐) are factors of p𝒙^𝟐+ 5x + r, prove that p = r.
Without actual division, prove that 𝟐𝒙^𝟒 − 𝟓𝒙^𝟑 + 𝟐𝒙^𝟐 − 𝒙 + 𝟐 is divisible by 𝒙^𝟐- 3x + 2.
What must be added to 𝟐𝒙^𝟒 − 𝟓𝒙^𝟑 + 𝟐𝒙^𝟐 − 𝒙 − 𝟑 so that the result is exactly divisible by (x-2)?
What must be subtracted from (𝒙^𝟒 + 𝟐𝒙^𝟑 − 𝟐𝒙^𝟐 + 𝟒𝒙 + 𝟔) so that the result is exactly divisible by
(𝒙^𝟐+2x – 3)?
Use factor theorem to prove that (x + a) is a factor of (𝒙^𝒏 + 𝒂^𝒏)for any odd positive integer n.
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