Polynomials and Remainder Theorem Practice Questions with Detailed Solutions – Level 1

Polynomials and Remainder Theorem Level 1 Practice: Factor Theorem Basics

This Level 1 set is meant for beginners starting polynomials, whether revising school algebra or building a base for aptitude exams. On Aptitude360 it covers the degree of a polynomial, remainders, the factor theorem, factorising quadratics and cubics, zeroes and their relation with coefficients, and the HCF and LCM of polynomials.

Topics in this polynomials Level 1 set

• Degree of a polynomial, including products and powers of polynomials
• Remainder on dividing by a linear expression using the Remainder Theorem
• Finding an unknown coefficient when a linear expression is a factor
• Factorising quadratics by splitting the middle term and cubics using one known factor
• Sum and product of zeroes and forming a quadratic from given zeroes
• HCF and LCM of simple polynomials

Key formulas for remainder and factor theorem questions

• Remainder Theorem: remainder on dividing \( p(x) \) by \( x-a \) is \( p(a) \)
• Factor Theorem: \( x-a \) is a factor if \( p(a)=0 \)
• Sum of zeroes of \( ax^{2}+bx+c \): \( \alpha+\beta=-\dfrac{b}{a} \)
• Product of zeroes of \( ax^{2}+bx+c \): \( \alpha\beta=\dfrac{c}{a} \)
• Product of zeroes of \( ax^{3}+bx^{2}+cx+d \): \( \alpha\beta\gamma=-\dfrac{d}{a} \)
• Quadratic from zeroes: \( x^{2}-(\alpha+\beta)x+\alpha\beta \)

How to revise polynomial basics

• Set the linear divisor equal to zero, then substitute the resulting value of x into the polynomial; the polynomial's value is the remainder.
• For a cubic, test small integer values that divide the constant term to find the first factor.
• Factorise both polynomials fully before finding their HCF or LCM.
• Remember that a polynomial of a given degree has at most that many zeroes.

Who this level suits

Students new to polynomials, including those preparing the algebra part of quantitative aptitude, who need firm basics before moving to the next level.