Polynomials and Remainder Theorem Practice Questions with Detailed Solutions – Level 3

Polynomials and Remainder Theorem Level 3: Advanced Practice Quiz

This Level 3 set on polynomials and the remainder theorem is for students who have finished the basics and want advanced algebra practice. It covers remainders by quadratic divisors, the factor theorem, relations between zeroes and coefficients, and HCF and LCM of polynomials. Aptitude360 gives a step-by-step solution for every question.

Topics covered in this Level 3 polynomials set

• Remainders on division by linear and quadratic divisors, including high powers of x
• Factor theorem to find unknown coefficients and complete factorisations
• Symmetric expressions of zeroes such as sums of squares, cubes and fourth powers
• Zeroes of cubics, including zeroes in arithmetic progression
• HCF and LCM of polynomials and finding a missing polynomial from them
• Degree of products and powers, and counting distinct real zeroes

Key formulas for polynomials and the remainder theorem

• Remainder theorem: remainder of \( p(x)\div(x-a) \) is \( p(a) \)
• Quadratic \( ax^{2}+bx+c \): \( \alpha+\beta=-\dfrac{b}{a},\ \alpha\beta=\dfrac{c}{a} \)
• Cubic \( ax^{3}+bx^{2}+cx+d \): \( \alpha+\beta+\gamma=-\dfrac{b}{a},\ \alpha\beta\gamma=-\dfrac{d}{a} \)
• Squares of zeroes: \( \alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta \)
• Cubes of zeroes: \( \alpha^{3}+\beta^{3}=(\alpha+\beta)^{3}-3\alpha\beta(\alpha+\beta) \)

How to revise remainder and zeroes problems

• For a quadratic divisor with distinct roots, write the remainder as a linear expression and substitute both roots
• For zeroes in arithmetic progression, take them as a minus d, a and a plus d so the middle zero comes from the sum
• Rewrite symmetric expressions using the sum and product of zeroes instead of solving for each zero
• For a divisor like x squared plus x plus one, use the fact that each complex cube root of unity cubes to one

Who should take this polynomials practice set

Aptitude and school algebra learners at the final stage of the topic, after the earlier levels, who want harder problems on remainders, zeroes and HCF and LCM.