Polynomials Class 10

This video focuses on mastering polynomials by exploring their key concepts, solving important questions, and learning effective tips and tricks for problem-solving. Viewers are encouraged to actively participate by practicing along with the content using pen and paper. Additional resources for practice can be found on Manoj Academy's website.

Understanding Zeros and Roots of Polynomials Zeros or roots of a polynomial are the values of x that make the polynomial equal to zero. For example, in the quadratic equation x² - 5x + 6 = 0, factorizing gives (x-3)(x-2)=0, so zeros are x=2 and x=3. These terms can be used interchangeably as they represent solutions where the polynomial equals zero.

Relation Between Zeros and Coefficients in Quadratic Polynomials In a quadratic polynomial ax²+bx+c with coefficients a≠0, there is an important relationship between its zeros (α & β) and coefficients: α+β=-b/a (sum), αβ=c/a (product). Using these relations allows rewriting any such quadratic as X²-(α+β)X+(αβ). This connection simplifies understanding polynomials' structure through their roots.

Finding and Verifying Zeros of a Quadratic Polynomial To find the zeros of the polynomial \(\sqrt{3}x^2 - 8x + 4\sqrt{3}\), set it to zero and use middle-term factorization. The factors are \((\sqrt{3}x-2)(x-2\sqrt{3})=0\), yielding roots \( x = 2/\sqrt{3}, x = 2"). Verification involves using formulas: sum of zeros equals \(-B/A) (here, resulting in "8/√"), while product equals C/A (resulting in "4"). Both calculations confirm consistency with coefficients.

Determining Coefficients from Given Roots Given roots for a quadratic equation, substitute them into its general form to derive equations involving coefficients. For example, substituting given values like X=−1 or X=-6 yields two linear equations that can be solved simultaneously determine unknowns precisely .

The problem involves finding the value of 'a' in a quadratic polynomial ax² - 6x - 6, given that the product of its zeros is 4. The concept hinges on using relationships between coefficients and roots: for any quadratic equation ax² + bx + c = 0, the product of zeros equals c/a. Applying this formula here gives (-6)/a = 4; solving it yields a as -3.