Master Polynomial Zeros in CBSE Class 10 Math

Stop Getting Stuck on Polynomial Problems! Here’s Your Complete Guide to Finding Zeros

If you’re a CBSE Class 10 student this seemingly tricky topic becomes super easy once you know the right approach. Let’s break down everything you need to know about finding zeros and understanding their relationship with coefficients.

What Are Polynomial Zeros?

Polynomial zeros (also called roots) are the values of x that make the polynomial equal to zero. For quadratic polynomials like 2x² + 14x + 20, there are typically two zeros. These aren’t just abstract concepts – they’re crucial for:

• Solving real-world problems
• Graphing polynomial functions
• Understanding parabola x-intercepts
• Acing your board exams!

The Step-by-Step Method: Middle Term Splitting

Let’s solve: Find the zeros of 2x² + 14x + 20

Step 1: Identify Your Strategy

We’ll use factorization by splitting the middle term. This method works great when the polynomial can be factored easily.

Step 2: Split the Middle Term

• Take the middle coefficient: 14
• Find two numbers that:
  • Add up to 14 (p + q = 14)
  • Multiply to give 40 (where did we get that 40 from? It’s the product of the coefficient of x², which is 2 and the constant term 20)

The magic numbers? 10 and 4

• 10 + 4 = 14 ✓
• 10 × 4 = 40 ✓

Step 3: Rewrite the Polynomial

2x² + 14x + 20 becomes: 2x² + 10x + 4x + 20

Step 4: Factor by Grouping

• Group 1: 2x² + 10x = 2x(x + 5)
• Group 2: 4x + 20 = 4(x + 5)
• Final form: (x + 5)(2x + 4)

Step 5: Find the Zeros

Set each factor to zero:

• x + 5 = 0 → x = -5
• 2x + 4 = 0 → x = -2

Answer: The zeros are -5 and -2

Quick Verification Method

Always verify your answer! Substitute x = -2: 2(-2)² + 14(-2) + 20 = 8 – 28 + 20 = 0 ✓

Substitute x = -5: 2(-5)² + 14(-5) + 20 = 50 – 70 + 20 = 0 ✓

Understanding the Coefficient Relationship

This is where many students get confused, but it’s actually straightforward:

For any quadratic ax² + bx + c:

Sum of zeros = -b/a

• Our polynomial: a = 2, b = 14, c = 20
• Sum = -14/2 = -7
• Check: (-5) + (-2) = -7 ✓

Product of zeros = c/a

• Product = 20/2 = 10
• Check: (-5) × (-2) = 10 ✓

This Method Works Every Time

The beauty of middle term splitting is its reliability. Once you master the pattern of finding two numbers that add to the middle coefficient and multiply to ac, you can solve any factorable quadratic polynomial.

Common Mistakes to Avoid

1. Sign errors: Pay attention to positive/negative signs
2. Wrong product: Remember it’s a × c, not just c
3. Incomplete factoring: Always check if factors can be simplified further
4. Skipping verification: Always plug your answers back in!

Practice Makes Perfect

The key to mastering polynomial zeros is practice. Try these techniques on different problems, and soon you’ll be solving them without breaking a sweat.

Ready for Your Next Challenge?

Now that you’ve conquered this fundamental concept, you’re ready to tackle more complex polynomial problems. Remember, every expert was once a beginner – keep practicing, and success will follow!

Need more help with CBSE Class 10 Math? Check out our complete video series covering every topic you need for board exam success!