# Polynomials | Revision Notes

# Polynomials

- An expression of the form p(x) = a
_{0}+ a_{1}x + a_{2}x^{2}+ a_{3}x^{3}+ ... + a_{n}x^{n}, where p(x), is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, a_{3}, ... a_{n } are real numbers and each power of x is a non-negative integer.

- The exponent of the highest degree term in a polynomial is known as its degree.
- A polynomial of degree 0 is called a
**constant polynomial**. - A polynomial of degree 1 is called a
**linear polynomial**. A linear polynomial is of the form p(x) = ax + b, where a ≠ 0. - A polynomial of degree 2 is called a
**quadratic polynomial**. A quadratic polynomial is of the form p(x) = ax^{2}+ bx + c, where a ≠ 0,. - A polynomial of degree 3 is called a
**cubic polynomial**. A cubic polynomial is of the form p(x) = ax^{3}+ bx^{2}+ cx + d, where a ≠ 0. - A polynomial of degree 4 is called a
**biquadratic polynomial**. A biquadratic polynomial is of the form p(x) = ax^{4}+ bx^{3}+ cx^{2}+ dx + e, where a ≠ 0,. - If p(x) is a polynomial in x and if α is any real number, then the value obtained by putting x = α in p(x) is called the value of p(x) at x = α . The value of p(x) at x = α is denoted by p(α) .
- A real number α is called a zero of the polynomial p(x), if p(α) = 0.
- A polynomial of degree n can have at most n real zeroes.
- Geometrically the zeroes of a polynomial p(x) are the x-coordinates of the points, where the graph of p(α) = 0. intersects x-axis.
- Zero of the linear polynomial ax + b is

- If α and β are the zeroes of a quadratic polynomial p(x) = ax
^{2}+ bx + c, a ≠ 0,, then

- If α , β and γ are the zeroes of a cubic polynomial p(x) = ax3 + bx2 + cx + d, a ≠ 0, then

- A quadratic polynomial whose zeroes are α , β is given by p(x) = x
^{2}– (α + β)x + αβ = x^{2}– (sum of the zeroes) x + product of the zeroes. - A cubic polynomial whose zeroes are α, β , γ is given by p(x) = x
^{3}– (α + β + γ) x^{2}+ (αβ + βγ + γα)x + αβγ = x^{3}– (sum of the zeroes) x^{2}+ (sum of the products of the zeroes taken two at a time) x – product of the zeroes. - The division algorithm states that given any polynomial p(x) and any non-zero polynomial g(x), there are polynomial q(x) and r(x) such that p(x) = g(x)q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

# Important Questions

- For what value of k, (–4) is a zero of the polynomial x
^{2}– x – (2k + 2)? - If the zeroes of the polynomial x
^{3}– 3x^{2}+ x + 1 are a – b, a and a + b, fnd the values of a and b. - Find a quadratic polynomial whose zeroes are 3 + √5 and 3–√5.
- α, β are the roots of the quadratic polynomial p(x) = x
^{2}– (k – 6) x + (2k + 1). Find the value of k, if α + β = αβ . - On dividing x
^{3}– 3x^{2}+ x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4 respectively. Find g(x). - Find the zeroes of 4√5x
^{2}+ 17x + 3√5 and verify the relation between the zeroes and coeffcients of the polynomial. - Find the zeroes of the quadratic polynomial p(x) = x
^{2}– (√3 +1)x + 3 and verify the relationship between the zeroes and its coeffcients. - If α and β are the zeroes of the quadraticpolynomial f(x) = x
^{2}– px + q, prove that

- If α and β are the zeroes of the quadraticpolynomial p(s) = 3s
^{2}– 6s + 4, fnd the value of

- If α and β are zeroes of the quadratic polynomial f(x) = kx
^{2}+ 4x + 4 such that α^{2}+ β^{2}= 24, find the value of k. - If one zero of the quadratic polynomial f(x) = 4x
^{2}– 8kx – 9 is negative of the other, find the value of k. - If the sum of the zeroes of the quadratic polynomial f(t) = kt
^{2}+ 2t + 3k is equal to their product, find the value of k. - If the square of the difference of the zeroes of the quadratic polynomial f(x) = x
^{2}+ px + 45 is equal to 144, find the value of p. - If one zero of the polynomial (a
^{2}+ 9)x^{2}+ 13x + 6a is reciprocal of the other, find the value of a. - If α, β, γ are zeroes of the polynomial 6x
^{3}+ 3x^{2}– 5x + 1, then fnd the value of α^{–1}+ β^{–1}+ γ^{–1}. - Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time and product of its zeroes as 3, –1 and –3 respectively.
- If α, β are the zeroes of the polynomial 21y
^{2}– y – 2, find a quadratic polynomial whose zeroes are 2α and 2β. - If α, β are the zeroes of the polynomial 6y
^{2}– 7y + 2, find a quadratic polynomial whose zeroes are - If α and β are the zeroes of the quadratic polynomial f(x) = x
^{2}+ px + q, form a polynomial whose zeroes are (α + β)^{2}and (α – β)^{2}. - If α and β are the zeroes of the polynomial f(x) = x
^{2}– 2x + 3, find a polynomial whose zeroes are α + 2 and α + β. - Divide 30x
^{4}+ 11x^{3}– 82x^{2}– 12x + 48 by (3x^{2}+ 2x – 4) and verify the result by division algorithm. - If α and β are zeroes of the quadratic polynomial x
^{2}– 6x + a; fnd the value of a if 3α + 2β = 20. - Find the value of p for which the polynomial x
^{3}+ 4x^{2}– px + 8 is exactly divisible by x – 2. - If the remainder on division of x
^{3}+ 2x^{2}+ kx + 3 by x – 3 is 21, fnd the quotient and the value of k. Hence, fnd the zeroes of the cubic polynomial x^{3}+ 2x^{2}+ kx – 18. - If the polynomial x
^{4}– 6x^{3}+ 16x^{2}– 25x + 10 is divided by another polynomial x^{2}– 2x + k, the remainder comes out to be (x + a), fnd the values of k and a. - Find all other zeroes of the polynomial p(x) = 2x
^{3}+ 3x^{2}– 11x – 6, if one of its zero is –3. - Find all the zeroes of the polynomial 2x
^{4}– 3x^{3}– 5x^{2}+ 9x – 3, it being given that two of its zeros are √3 and –√3. - Find all the zeroes of the polynomial 2x
^{4}– 10x^{3}+ 5x^{2}+ 15x – 12, if it is given that two of its zeroes are . - Find all the zeroes of the polynomial x
^{4}+ x^{3}– 34x^{2}– 4x + 120, if two of its zeroes are 2 and –2. - If two zeroes of p(x) = x
^{4}– 6x^{3}– 26x^{2}+ 138x – 35 are 2 ± √3 , fnd the other zeroes. - If the polynomial 6x
^{4}+ 8x^{3}– 5x^{2}+ ax + b is exactly divisible by the polynomial 2x^{2}– 5, then fnd the values of a and b. - What must be added to the polynomial P(x) = 5x
^{4}+ 6x^{3}– 13x^{2}– 44x + 7 so that the resulting polynomial is exactly divisible by the polynomial Q(x) = x^{2}+ 4x + 3 and the degree of the polynomial to be added must be less than degree of the polynomial Q(x). - What must be subtracted from the polynomial f(x) = x
^{4}+ 2x^{3}– 13x^{2}– 12x + 21 so that the resulting polynomial is exactly divisible by g(x) = x^{2}– 4x + 3?