Polynomials: Finding Values and Zeros (NCERT Ex 2.2)

What is the Value of a Polynomial?

Finding the value of a polynomial is a straightforward process of substitution. If you have a polynomial, say p(x), its value at x = k is found by replacing every instance of x in the expression with the number k and then simplifying. The notation we use is p(k).

Let's take the polynomial p(x) = 5x² - 3x + 7.

What is its value at x = 1? We need to find p(1).

1. Substitute: Replace every x with 1:

p(1) = 5(1)² - 3(1) + 7

1. Calculate: Follow the order of operations (BODMAS/PEMDAS). First, the exponent:

p(1) = 5(1) - 3(1) + 7

1. Simplify: Now, multiply and then add/subtract:

p(1) = 5 - 3 + 7 = 2 + 7 = 9

So, the value of the polynomial p(x) = 5x² - 3x + 7 at x = 1 is 9.

What about at x = 0?
p(0) = 5(0)² - 3(0) + 7 = 0 - 0 + 7 = 7.

This simple skill is the first step towards understanding how polynomials behave.

Key Concept: Zeros of a Polynomial

Zero of a Polynomial

A real number 'k' is called a zero of the polynomial p(x) if p(k) = 0. In other words, a zero is a specific input value for the variable that makes the entire polynomial's output equal to zero.

Root of a Polynomial Equation

The 'zeros' of a polynomial p(x) are the same as the 'roots' of the polynomial equation p(x) = 0. These terms are often used interchangeably.

How to Find and Verify Zeros

1. Verifying if a Number is a Zero — To check if a given number, say k, is a zero of a polynomial p(x), you simply calculate the value of the polynomial at x = k. If p(k) equals 0, then k is a zero. If p(k) is any other number, k is not a zero. For example, to verify if x = 2 is a zero of p(x) = x - 2, we find p(2) = 2 - 2 = 0. Since the result is 0, x=2 is a zero.
2. Finding the Zero of a Linear Polynomial — For a linear polynomial (degree 1), like p(x) = ax + b, there is a simple method to find its zero. Set the entire polynomial expression equal to zero and solve the resulting equation for x. ax + b = 0 ax = -b x = -b/a This value, x = -b/a, is the one and only zero of the linear polynomial ax + b.

Common Mistakes to Avoid

1. Sign Errors During Substitution: When you substitute a negative number, always use parentheses to avoid mistakes. For p(x) = x² - 2x at x = -3, write p(-3) = (-3)² - 2(-3). This becomes 9 - (-6), which is 9 + 6 = 15. A common error is writing -3² which can be misinterpreted as -9.

2. Confusing 'Zero of a Polynomial' with the number 0: Remember, a 'zero of a polynomial' is the input value of x that makes the polynomial's value 0. For p(x) = x - 5, the zero is x=5, not x=0. Finding p(0) gives the y-intercept, not the zero (unless the zero itself is 0).

3. Forgetting to Set p(x)=0: When asked to find the zero of a linear polynomial like p(x) = 2x + 6, you must set 2x + 6 = 0 and solve. A common mistake is just substituting a random number. The goal is to find the specific x that makes the expression zero.

Practice Questions with Solutions

• Q: Find the value of the polynomial p(x) = 5x - 4x² + 3 at x = 2. A: Step 1: Write down the polynomial: p(x) = 5x - 4x² + 3. Step 2: Substitute x = 2 into the expression. p(2) = 5(2) - 4(2)² + 3 Step 3: Calculate the terms, following order of operations (exponents first). p(2) = 10 - 4(4) + 3 p(2) = 10 - 16 + 3 Step 4: Simplify by addition and subtraction. p(2) = -6 + 3 = -3 Final answer: The value of the polynomial at x = 2 is -3.
• Q: Verify whether x = 2 and x = 3 are zeros of the polynomial p(x) = x² - 5x + 6. A: Step 1: To check for x = 2, we calculate p(2). p(2) = (2)² - 5(2) + 6 p(2) = 4 - 10 + 6 p(2) = -6 + 6 = 0 Since p(2) = 0, x = 2 is a zero of the polynomial. Step 2: To check for x = 3, we calculate p(3). p(3) = (3)² - 5(3) + 6 p(3) = 9 - 15 + 6 p(3) = -6 + 6 = 0 Since p(3) = 0, x = 3 is also a zero of the polynomial. Final answer: Yes, both x = 2 and x = 3 are zeros of p(x) = x² - 5x + 6.
• Q: Find the zero of the linear polynomial p(t) = 4t + 8. A: Step 1: To find the zero, we must set the polynomial expression equal to zero. p(t) = 0 4t + 8 = 0 Step 2: Solve the linear equation for the variable t. 4t = -8 t = -8 / 4 t = -2 Final answer: The zero of the polynomial p(t) = 4t + 8 is t = -2.
• Q: Is 0 a zero of the polynomial p(x) = 6x³ - 2x? What about the polynomial q(x) = x + 4? A: Step 1: For the first polynomial, p(x) = 6x³ - 2x, let's find the value at x=0. p(0) = 6(0)³ - 2(0) = 6(0) - 0 = 0 - 0 = 0. Since p(0) = 0, yes, 0 is a zero of p(x). Step 2: For the second polynomial, q(x) = x + 4, let's find the value at x=0. q(0) = 0 + 4 = 4. Since q(0) is not equal to 0, 0 is not a zero of q(x). Final answer: 0 is a zero of p(x) = 6x³ - 2x but not of q(x) = x + 4.

Frequently Asked Questions