NCERT Class 10 Maths Chapter 2 Exercise 2.3: Polynomials

The Division Algorithm for Polynomials Explained

The Division Algorithm states that for any two polynomials $p(x)$ and $g(x)$, where $g(x) \neq 0$, we can find unique polynomials $q(x)$ and $r(x)$ such that:
$p(x) = g(x) \cdot q(x) + r(x)$

Here, $p(x)$ is the dividend, $g(x)$ is the divisor, $q(x)$ is the quotient, and $r(x)$ is the remainder. An essential rule of this algorithm is that either the remainder $r(x) = 0$ or the degree of $r(x) <$ degree of $g(x)$. If the remainder $r(x) = 0$, it implies that $g(x)$ is a factor of $p(x)$. This relation is extremely helpful when we need to factorize higher-degree polynomials.

Step-by-Step Method for Polynomial Long Division

1. Standardize the Polynomials — Arrange the terms of both the dividend $p(x)$ and the divisor $g(x)$ in decreasing order of their exponents (standard form). If any intermediate power of $x$ is missing, write it with a coefficient of 0 (e.g., write $x^3 - 1$ as $x^3 + 0x^2 + 0x - 1$) to keep columns aligned.
2. Calculate the First Quotient Term — Divide the term with the highest degree in the dividend by the term with the highest degree in the divisor. This result becomes the first term of your quotient $q(x)$.
3. Multiply and Subtract — Multiply the entire divisor $g(x)$ by the newly found quotient term. Align like terms under the dividend and subtract this product from the dividend to obtain a new temporary remainder.
4. Repeat Until Termination — Treat the temporary remainder as your new dividend. Repeat steps 2 and 3 until the degree of your remainder is strictly less than the degree of the divisor $g(x)$.

Avoid Common Mistakes in Board Exams

1. Sign Mistakes During Subtraction: This is the absolute number-one source of errors! When subtracting the multiplied product from the dividend, remember to change the signs of every term in the product.
2. Ordering Terms: Never skip rearranging terms in descending order of power. Dividing with disordered polynomials leads to algebraic dead-ends.
3. Finding All Roots: In 'find all other zeroes' questions, when given roots like $\alpha$ and $\beta$, the divisor is $(x-\alpha)(x-\beta)$. Perform division carefully; your remainder must be zero. If it isn't, you made a calculation error.

Practice Questions with Solutions

• Q: Divide $p(x) = x^3 - 3x^2 + 5x - 3$ by $g(x) = x^2 - 2$ and find the quotient and remainder. A: Step 1: Check standard form. Both polynomials are arranged in descending powers of $x$. Step 2: Divide the first term of dividend ($x^3$) by first term of divisor ($x^2$). Quotient term is $x$. Step 3: Multiply divisor by $x$: $x(x^2 - 2) = x^3 - 2x$. Subtract this from $p(x)$ to get the new remainder: $-3x^2 + 7x - 3$. Step 4: Divide first term of new remainder ($-3x^2$) by $x^2$ to get $-3$. Multiply divisor by $-3$: $-3(x^2 - 2) = -3x^2 + 6$. Subtract to get final remainder: $7x - 9$. Final answer: Quotient $q(x) = x - 3$, Remainder $r(x) = 7x - 9$.
• Q: Find all the zeroes of $3x^4 + 6x^3 - 2x^2 - 10x - 5$, if two of its zeroes are $\sqrt{\frac{5}{3}}$ and $-\sqrt{\frac{5}{3}}$. A: Step 1: Since $\sqrt{5/3}$ and $-\sqrt{5/3}$ are zeroes, $(x - \sqrt{5/3})(x + \sqrt{5/3}) = x^2 - 5/3$ is a factor. Multiplying by 3, $3x^2 - 5$ is also a factor. Step 2: Divide $3x^4 + 6x^3 - 2x^2 - 10x - 5$ by $3x^2 - 5$ using polynomial division. Step 3: Division yields quotient $x^2 + 2x + 1$ and remainder $0$. Step 4: Factorize the quotient: $x^2 + 2x + 1 = (x+1)^2 = 0 \implies x = -1, -1$. Final answer: The remaining zeroes are $-1$ and $-1$.
• Q: On dividing $x^3 - 3x^2 + x + 2$ by a polynomial $g(x)$, the quotient and remainder were $x - 2$ and $-2x + 4$, respectively. Find $g(x)$. A: Step 1: Use Division Algorithm: $p(x) = g(x) \cdot q(x) + r(x)$. Step 2: Substitute given expressions: $x^3 - 3x^2 + x + 2 = g(x)(x-2) + (-2x + 4)$. Step 3: Rearrange to solve for $g(x)$: $g(x)(x-2) = (x^3 - 3x^2 + x + 2) - (-2x + 4) = x^3 - 3x^2 + 3x - 2$. Step 4: Divide $x^3 - 3x^2 + 3x - 2$ by $x - 2$ to find $g(x)$. Long division yields quotient $x^2 - x + 1$ with 0 remainder. Final answer: $g(x) = x^2 - x + 1$.
• Q: Check whether $x^2 + 3x + 1$ is a factor of $3x^4 + 5x^3 - 7x^2 + 2x + 2$. A: Step 1: Perform long division of $p(x) = 3x^4 + 5x^3 - 7x^2 + 2x + 2$ by $g(x) = x^2 + 3x + 1$. Step 2: Divide $3x^4$ by $x^2$ to get quotient term $3x^2$. Subtract product to get new remainder: $-4x^3 - 10x^2 + 2x + 2$. Step 3: Divide $-4x^3$ by $x^2$ to get quotient term $-4x$. Subtract product to get new remainder: $2x^2 + 6x + 2$. Step 4: Divide $2x^2$ by $x^2$ to get quotient term $2$. Subtract product to get remainder $0$. Final answer: Since the remainder is $0$, $x^2 + 3x + 1$ is a factor of the given polynomial.

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