Factors and Multiples of Polynomials MCQs Quiz | Class 9

Class: IX, Subject: Mathematics (Code 041), Unit II: Algebra. This quiz covers the Factor concept, the idea of division in polynomials, and multiples. Test your knowledge of finding factors, using the Factor Theorem, and understanding polynomial division. Submit your answers to see your score and download the PDF solution sheet.

Understanding Factors and Multiples of Polynomials

In Algebra, particularly for Class IX, understanding how polynomials interact through division, factors, and multiples is crucial. Just as integers have factors (numbers that divide them exactly) and multiples, polynomials share similar properties defined by algebraic rules.

1. The Factor Concept

A polynomial g(x) is said to be a factor of another polynomial p(x) if g(x) divides p(x) completely, leaving a remainder of zero. This is analogous to saying 3 is a factor of 12 because 12 divided by 3 leaves no remainder.

  • Factor Theorem: If p(x) is a polynomial of degree n greater than or equal to 1 and ‘a’ is any real number, then (x – a) is a factor of p(x) if p(a) = 0.
  • Conversely, if (x – a) is a factor of p(x), then p(a) = 0.

2. The Idea of Division

When we divide a polynomial p(x) (dividend) by a non-zero polynomial g(x) (divisor), we get a quotient q(x) and a remainder r(x). This is expressed by the Division Algorithm:

Dividend = Divisor x Quotient + Remainder
p(x) = g(x)q(x) + r(x)

The degree of the remainder r(x) is always strictly less than the degree of the divisor g(x), or r(x) is zero.

3. Multiples of Polynomials

If a polynomial p(x) can be written as the product of two or more polynomials, then p(x) is called a multiple of each of those factors. For example, since x^2 – 4 = (x – 2)(x + 2), the polynomial x^2 – 4 is a multiple of both (x – 2) and (x + 2).

Key Points to Remember

  • To check if (x – a) is a factor of p(x), simply calculate p(a). If the result is 0, it is a factor.
  • To find the remainder when p(x) is divided by (x – a), calculate p(a). This is the Remainder Theorem.
  • Splitting the middle term is a common method to factorize quadratic polynomials of the form ax^2 + bx + c.

Practice Questions

  1. Find the value of k if (x – 1) is a factor of 2x^2 + kx + root(2).
  2. Check whether 7 + 3x is a factor of 3x^3 + 7x.
  3. Factorize: 12x^2 – 7x + 1.
  4. Use the Factor Theorem to determine if (x + 2) is a factor of x^3 + 3x^2 + 3x + 1.
  5. Find the remainder when x^4 + x^3 – 2x^2 + x + 1 is divided by x – 1.