Polynomials and Remainder Theorem Practice Questions with Detailed Solutions – Level 2

Polynomials and Remainder Theorem Level 2 Questions: Zeroes and Factors

Level 2 of the Polynomials and Remainder Theorem practice series on Aptitude360 moves past direct substitution into multi-step algebra. It tests remainders on dividing by a linear term with a leading coefficient, conditions that fix two unknown coefficients, symmetric expressions in the zeroes of quadratics and cubics, and HCF and LCM relations.

What the Level 2 polynomials questions test

• Remainder on dividing by a linear divisor whose leading coefficient is not one
• Finding one or two unknown coefficients from given factors or remainders
• Symmetric functions of zeroes such as sum of squares, sum of cubes and square of the difference
• Zeroes of a cubic in arithmetic progression and zeroes in a given ratio
• Complete factorisation of cubics and quadratics with surd coefficients
• Counting distinct real zeroes and using the HCF and LCM relation

Key formulas and rules for zeroes and remainders

• Divisor \( ax-b \): remainder is \( p\left(\dfrac{b}{a}\right) \)
• Sum of squares: \( \alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta \)
• Square of difference: \( (\alpha-\beta)^{2}=(\alpha+\beta)^{2}-4\alpha\beta \)
• Sum of cubes: \( \alpha^{3}+\beta^{3}=(\alpha+\beta)^{3}-3\alpha\beta(\alpha+\beta) \)
• Cubic \( ax^{3}+bx^{2}+cx+d \): \( \alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{c}{a} \)
• Same cubic: \( \dfrac{1}{\alpha}+\dfrac{1}{\beta}+\dfrac{1}{\gamma}=-\dfrac{c}{d} \)

Preparation tips for this level

• Express any symmetric expression in terms of the sum and product of zeroes before substituting values.
• For zeroes in arithmetic progression, take them as a minus d, a and a plus d so the sum gives the middle zero at once.
• When two factors are given, write two equations from the Factor Theorem and solve them together.
• A squared factor gives a repeated zero, and a square plus a positive number gives no real zero.

Who should practise this set

Aptitude aspirants who are comfortable with the first level of polynomial basics and are ready for medium-level problems on zeroes and factors.