Solving Class 10 Polynomial Questions: Techniques and Methods
Solving Class 10 Polynomial Questions: Techniques and Methods
Introduction
Solving polynomial questions is a fundamental skill in algebra, especially for class 10 students. Whether it is finding the roots of a polynomial or performing polynomial division, mastering these techniques is essential for success in math exams and competitions. This article focuses on various methods such as long division, the discriminant method, and simple factorization to solve these types of problems.Long Division Method - Polynomials
When faced with a complex polynomial expression, the long division method can be a powerful tool. For instance, if you need to divide a polynomial by a linear expression, you can use the long division method to find the quotient and the remainder. Here’s a step-by-step guide on how to apply the long division method to solve polynomial questions. Example: Divide (6x^3 sqrt{2}x^2 - 1 - 4sqrt{2}) by (x - sqrt{2}). Step 1:Set up the long division with the dividend (6x^3 sqrt{2}x^2 - 1 - 4sqrt{2}) and the divisor (x - sqrt{2}).
Step 2:Divide the leading term of the dividend by the leading term of the divisor to find the first quotient term: (6x^3 / x 6x^2).
Step 3:Multiply the entire divisor (x - sqrt{2}) by the quotient term (6x^2) and subtract the result from the dividend:
[(6x^2)(x - sqrt{2}) 6x^3 - 6sqrt{2}x^2.] Step 4:Subtract the result from the original polynomial:
[(6x^3 sqrt{2}x^2 - 1 - 4sqrt{2}) - (6x^3 - 6sqrt{2}x^2) 7sqrt{2}x^2 - 1 - 4sqrt{2}.] Step 5:Repeat the process to continue division:
Step 6:Divide the new leading term (7sqrt{2}x^2) by the divisor (x - sqrt{2}):
[7sqrt{2}x^2 / x 7sqrt{2}x.] Step 7:Multiply and subtract:
[(7sqrt{2}x)(x - sqrt{2}) 7sqrt{2}x^2 - 14.] Step 8:Subtract the result from the current polynomial:
[(7sqrt{2}x^2 - 1 - 4sqrt{2}) - (7sqrt{2}x^2 - 14) 13 - 4sqrt{2}.] Step 9:Divide the new leading term (13 - 4sqrt{2}) by the divisor (x - sqrt{2}):
[(13 - 4sqrt{2}) / (x - sqrt{2}) 4 sqrt{2}.] Thus, the quotient is (6x^2 7sqrt{2}x 4 sqrt{2}) and the remainder is 0. Therefore, the polynomial can be written as: [6x^3 sqrt{2}x^2 - 1 - 4sqrt{2} (x - sqrt{2})(6x^2 7sqrt{2}x 4 sqrt{2})]Discriminant Method
The discriminant method is another useful technique for finding the roots of a quadratic equation. The discriminant (D) of a quadratic equation (ax^2 bx c 0) is given by (D b^2 - 4ac). If (D > 0), the equation has two distinct real roots; if (D 0), it has a double root; and if (D Example: The polynomial (6x^3 sqrt{2}x^2 - 1 - 4sqrt{2}) has (sqrt{2}) as one of its zeroes. Hence, the polynomial can be written as the product of three linear polynomials. Step 1:Given that (sqrt{2}) is a zero, the polynomial can be written as ((x - sqrt{2})(ax^2 bx c)).
Step 2:To find (a), (b), and (c), we perform polynomial division:
Step 3:Dividing (6x^3 sqrt{2}x^2 - 1 - 4sqrt{2}) by (x - sqrt{2}) gives us the quotient (6x^2 7sqrt{2}x 4 sqrt{2}).
Step 4:Now, use the discriminant method to find the roots of (6x^2 7sqrt{2}x 4 sqrt{2}) using the quadratic formula:
[x frac{-b pm sqrt{D}}{2a}] Step 5:For the equation (6x^2 7sqrt{2}x 4 sqrt{2} 0), we have (a 6), (b 7sqrt{2}), and (c 4 sqrt{2}).
Step 6:Calculate the discriminant (D):
[D (7sqrt{2})^2 - 4 cdot 6 cdot (4 sqrt{2}) 98 - 24 cdot (4 sqrt{2}) 98 - 96 - 24sqrt{2} 2 - 24sqrt{2}] Step 7:Note: Since (D Step 8:
So, the roots of (6x^2 7sqrt{2}x 4 sqrt{2} 0) are complex numbers.
Simple Factorization Method
Factorization is another method to solve polynomial questions. Breaking down the polynomial into simpler, manageable factors can be very helpful. Let’s use this method to find the roots of the quadratic equation (6x^2 7sqrt{2}x 4 sqrt{2} 0). Example: The polynomial (6x^2 7sqrt{2}x 4 sqrt{2} 0). Step 1:Identify the sum and product of the roots using the coefficients. For a quadratic equation (ax^2 bx c 0), the sum of the roots is (-b/a) and the product of the roots is (c/a).
Step 2:For (6x^2 7sqrt{2}x 4 sqrt{2} 0), the sum of the roots is (-7sqrt{2}/6) and the product is ((4 sqrt{2})/6).
Step 3:Factorize the quadratic equation by splitting the middle term:
[6x^2 7sqrt{2}x 4 sqrt{2} 6x^2 3sqrt{2}x 4sqrt{2}x 4 sqrt{2} (3sqrt{2}x 4)(sqrt{2}x 1) sqrt{2}x 1 0] Step 4:Thus, the polynomial can be factored as ((3sqrt{2}x 4)(sqrt{2}x 1) 0).
Step 5:Solve for (x) using each factor:
[3sqrt{2}x 4 0 implies x -frac{4}{3sqrt{2}} -frac{2sqrt{2}}{3}] [ sqrt{2}x 1 0 implies x -frac{1}{sqrt{2}} -frac{sqrt{2}}{2}] Therefore, the roots of the quadratic equation are (x -frac{2sqrt{2}}{3}) and (x -frac{sqrt{2}}{2}).