Polynomials Solved Examples (Class 9 Maths)
Polynomials are algebraic expressions with variables and coefficients. Understanding polynomial operations, factorization, and the remainder theorem helps
TL;DR: Polynomials are algebraic expressions with variables and coefficients. Understanding polynomial operations, factorization, and the remainder theorem h…
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Polynomials are algebraic expressions with variables and coefficients. Understanding polynomial operations, factorization, and the remainder theorem helps
Polynomials — Solved Numerical Examples (Step by Step)
Example 1: Find the degree of the polynomial 3x^4 + 2x^3 - 5x + 7.
Solution: A polynomial's degree is the highest power of the variable in the expression.
In the polynomial 3x^4 + 2x^3 - 5x + 7:
The term 3x^4 has the highest power.
The exponent of x is 4.
Degree of the polynomial = 4
Example 2: Verify that x = 1 is a zero of the polynomial p(x) = x^3 - 2x^2 + 3x - 2.
Solution: A number 'a' is a zero of polynomial p(x) if p(a) = 0.
Substitute x = 1 in p(x) = x^3 - 2x^2 + 3x - 2:
p(1) = (1)^3 - 2(1)^2 + 3(1) - 2
p(1) = 1 - 2 + 3 - 2
p(1) = 0
Since p(1) = 0, x = 1 is a zero of the polynomial.
Example 3: Factorize the polynomial x^3 - 8.
Solution: x^3 - 8 = x^3 - 2^3
This is a difference of cubes. Using the formula:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
Example 4: Find the quotient and remainder when x^3 + 2x^2 - 5x - 6 is divided by x - 2.
Solution: Using synthetic division or long division:
Divide x^3 + 2x^2 - 5x - 6 by x - 2:
Coefficients: 1, 2, -5, -6
Divisor: x - 2 (so use 2)
Synthetic division:
2 | 1 2 -5 -6
| _ 2 8 6
1 4 3 0
Quotient = x^2 + 4x + 3
Remainder = 0
Example 5: Evaluate p(x) = 2x^2 - 3x + 5 at x = -2.
Solution: Substitute x = -2 in p(x) = 2x^2 - 3x + 5:
p(-2) = 2(-2)^2 - 3(-2) + 5
p(-2) = 2(4) + 6 + 5
p(-2) = 8 + 6 + 5
p(-2) = 19
Example 6: Factorize x^2 + 7x + 12.
Solution: We need to find two numbers that multiply to 12 and add to 7.
Those numbers are 3 and 4.
x^2 + 7x + 12 = (x + 3)(x + 4)
Example 7: Use the remainder theorem to find the remainder when p(x) = x^4 - 2x^3 + 3x - 5 is divided by x - 1.
Solution: By the remainder theorem, when p(x) is divided by (x - a), the remainder is p(a).
Here, a = 1. So the remainder is p(1).
p(1) = (1)^4 - 2(1)^3 + 3(1) - 5
p(1) = 1 - 2 + 3 - 5
p(1) = -3
Therefore, the remainder is -3.
Tips
- The degree of a polynomial is the highest power of the variable. Always look at all terms to find the one with the highest exponent.
- To check if a number is a zero of a polynomial, substitute it and see if the result is 0. If p(a) = 0, then a is a zero and (x - a) is a factor.
- Use the factor theorem and remainder theorem together: if p(a) = 0, then (x - a) is a factor of p(x). If p(a) = r (not zero), then r is the remainder when divided by (x - a).
Frequently Asked Questions
What is the difference between the degree and the number of terms in a polynomial?
The degree is the highest power of the variable, not the number of terms. For example, x^5 + 2x^3 - x + 1 has degree 5 (highest power is 5) but has 4 terms. A polynomial can have any number of terms, but degree refers to one specific value.
How do I use the factor theorem?
The factor theorem states that (x - a) is a factor of polynomial p(x) if and only if p(a) = 0. So if you find that p(a) = 0 for some value a, then you know (x - a) is a factor. This helps in factorization and finding zeros.
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